System, Method and Apparatus for Improving Accuracy of RF Transmission Models for Selected Portions of an RF Transmission Path

ABSTRACT

Systems and methods for determining an RF transmission line model for an RF transmission system includes generating a baseline RF transmission line model characterizing the RF transmission system. A plasma RF voltage, RF current, RF power and/or a corresponding RF induced DC bias voltage is calculated from the baseline RF transmission line model. An end module including the electrostatic chuck, a plasma and an RF return path is added to the baseline RF transmission line model to create one or more revised RF transmission line models. A revised plasma RF voltage, a revised plasma RF current, a revised plasma RF power and/or a corresponding revised RF induced DC bias voltage is calculated from each of the revised baseline RF transmission line models. The revised RF transmission line models are scored to identify a best fitting revised RF transmission line model as a complete RF transmission line model.

FIELD

The present embodiments relate to using RF modeling of a RF transmissionsystem to determine RF model of selected stages in the RF transmissionsystem in a plasma processing system.

BACKGROUND

In a plasma-based system, plasma is generated within a plasma chamber toperform various operations, e.g., etching, cleaning, depositing, etc.,on a wafer. The plasma is monitored and controlled to controlperformance of the various operations. For example, the plasma ismonitored by monitoring a voltage of the plasma and is controlled bycontrolling an amount of radio frequency (RF) power supplied to theplasma chamber.

However, the use of voltage to monitor and control the performance ofthe operations may not provide satisfactory results. Moreover, themonitoring of voltage may be an expensive and time consuming operation.

It is in this context that embodiments described in the presentdisclosure arise.

SUMMARY

Embodiments of the disclosure provide apparatus, methods and computerprograms for using modeling to identify locations of faults within an RFtransmission system in a plasma system. It should be appreciated thatthe present embodiments can be implemented in numerous ways, e.g., aprocess, an apparatus, a system, a piece of hardware, or a method on acomputer-readable medium. Several embodiments are described below.

One embodiment provides a method for determining an RF transmission linemodel for an RF transmission system includes generating a baseline RFtransmission line model characterizing the RF transmission system. Aplasma RF voltage, RF current, RF power and/or a corresponding RFinduced DC bias voltage is calculated from the baseline RF transmissionline model. An end module including the electrostatic chuck, a plasmaand an RF return path is added to the baseline RF transmission linemodel to create one or more revised RF transmission line models. Arevised plasma RF voltage, a revised plasma RF current, a revised plasmaRF power and/or a corresponding revised RF induced DC bias voltage iscalculated from each of the revised baseline RF transmission linemodels. The revised RF transmission line models are scored to identify abest fitting revised RF transmission line model as a complete RFtransmission line model.

Adding the end module to the baseline RF transmission line model caninclude selecting an initial range for each of equivalent serial andshunt RLC values representing the end module, dividing the selectedranges into a selected number of subdivisions to identify test valuescorresponding to each subdivision for each serial and shunt RLC values,testing each combination of the test values for each of the serial andshunt RLC values to identify a plurality of local minimums, recordingeach of the plurality of local minimums and selecting additional rangesfor each of equivalent serial and shunt RLC values representing the endmodule when a selected threshold number of local minimums is not met.

Selecting the initial range for each of equivalent serial and shunt RLCvalues representing the end module can include selecting a range ofequivalent serial and shunt RLC values including the equivalent serialand shunt RLC values in at least one other stage of the RF transmissionsystem. Alternatively or additionally, selecting the initial range foreach of equivalent serial and shunt RLC values representing the endmodule can include selecting random ranges of values for each equivalentserial and shunt RLC values.

Dividing the selected ranges into the selected number of subdivisions toidentify test values corresponding to each subdivision for each serialand shunt RLC values can include dividing each of the selected rangesinto an equal number of subdivisions for each of the ranges of theserial and shunt RLC values. Alternatively or additionally, dividing theselected ranges into the selected number of subdivisions to identifytest values corresponding to each subdivision for each serial and shuntRLC values can include dividing each of the selected ranges intonon-equal numbers of subdivisions for each of the ranges of the serialand shunt RLC values.

Selecting additional ranges for each of equivalent serial and shunt RLCvalues representing the end module can include selecting a multiple ofthe initial range for at least one of each of equivalent serial andshunt RLC values representing the end module. Alternatively oradditionally, selecting additional ranges for each of equivalent serialand shunt RLC values representing the end module includes selecting alarger range for at least one of each of equivalent serial and shunt RLCvalues representing the end module.

Adding an end module to the baseline RF transmission line model caninclude selecting an initial random value for each of equivalent serialand shunt RLC values representing the end module, testing eachcombination of the test values for each of the serial and shunt RLCvalues to identify a gradient, select new test values for each of theserial and shunt RLC values corresponding to the gradient toward a localminimum, recording the local minimum when a local minimum is identifiedand selecting additional random values for each of equivalent serial andshunt RLC values representing the end module when the threshold numberof local minimums is not met.

Scoring the revised RF transmission line models to identify the bestfitting revised RF transmission line model can include selecting atleast one bias model to test each revised RF transmission line model,testing each revised RF transmission line model with each of theselected at least one bias model, storing test results for each of theselected at least one bias model for each of the revised RF transmissionline model and scoring the stored test results according to accuracy ofthe revised plasma RF voltage (V′rf) and/or a revised plasma RF current(I′rf) and/or a revised plasma RF power (P′rf) and/or a correspondingrevised RF induced DC bias voltage corresponding to each of the at leastone revised baseline RF transmission line models.

Another embodiment provides a plasma system including a plasmaprocessing chamber, an RF transmission system coupled to an RF input ofthe plasma processing chamber, an RF generator having an output coupledto the RF transmission system and a controller coupled to the RFgenerator and the plasma processing chamber. The controller includinglogic on computer readable media being executable for adjusting at leastone dynamic first principal variable of an RF signal in a plasma in theplasma processing chamber according to a complete RF transmission linemodel of the RF transmission system to compensate for a differencebetween the plasma system and a second plasma system. The complete RFtransmission line model being determined by generating a baseline RFtransmission line model characterizing the RF transmission system. Thebaseline RF transmission line model having multiple stages ending at aninput to an electrostatic chuck. At least one of a plasma RF voltage(Vrf) and/or a plasma RF current (Irf) and/or a plasma RF power (Prf)and/or a corresponding RF induced DC bias voltage is calculated from thebaseline RF transmission line model. An end module is added to thebaseline RF transmission line model to create at least one revised RFtransmission line model, the end module including the electrostaticchuck, a plasma and an RF return path. At least one of a revised plasmaRF voltage (V′rf) and/or a revised plasma RF current (I′rf) and/or arevised plasma RF power (P′rf) and/or corresponding revised RF inducedDC bias voltage can be calculated from each of the at least one revisedbaseline RF transmission line models. Each of the revised RFtransmission line models can be scored to identify a best fittingrevised RF transmission line model and the best fitting revised RFtransmission line model can be recorded as a complete RF transmissionline model.

Advantages of the disclosed embodiments include improved control of aplasma processing chamber such as controlling multiple plasma processingchambers to operate substantially identically. The disclosed embodimentsalso provide the advantage of improved production methods for producingthe plasma processing chambers and the RF transmission systems thereofas the complete RF transmission model can be used to identify productionvariations in each plasma processing chamber.

Other aspects will become apparent from the following detaileddescription, taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments may best be understood by reference to the followingdescription taken in conjunction with the accompanying drawings.

FIG. 1 is a block diagram of a system for determining a variable at anoutput of an impedance matching model, at an output of a portion of aradio frequency (RF) transmission model, and at an output of anelectrostatic chuck (ESC) model, in accordance with an embodimentdescribed in the present disclosure.

FIG. 2 is a flowchart of a method for determining a complex voltage andcurrent at the output of the RF transmission model portion, inaccordance with an embodiment described in the present disclosure.

FIG. 3A is a block diagram of a system used to illustrate an impedancematching circuit, in accordance with an embodiment described in thepresent disclosure.

FIG. 3B is a circuit diagram of an impedance matching model, inaccordance with an embodiment described in the present disclosure.

FIG. 4 is a diagram of a system used to illustrate an RF transmissionline, in accordance with an embodiment described in the presentdisclosure.

FIG. 5A is a block diagram of a system used to illustrate a circuitmodel of the RF transmission line, in accordance with an embodimentdescribed in the present disclosure.

FIG. 5B is a diagram of an electrical circuit used to illustrate atunnel and strap model of the RF transmission model, in accordance withan embodiment described in the present disclosure.

FIG. 5C is a diagram of an electrical circuit used to illustrate atunnel and strap model, in accordance with an embodiment described inthe present disclosure.

FIG. 6 is a diagram of an electrical circuit used to illustrate acylinder and ESC model, in accordance with an embodiment described inthe present disclosure.

FIG. 7 is a block diagram of a plasma system that includes filters usedto determine the variable, in accordance with an embodiment described inthe present disclosure.

FIG. 8A is a diagram of a system used to illustrate a model of thefilters to improve an accuracy of the variable, in accordance with anembodiment described in the present disclosure.

FIG. 8B is a diagram of a system used to illustrate a model of thefilters, in accordance with an embodiment described in the presentdisclosure.

FIG. 9 is a block diagram of a system for using a current and voltageprobe to measure the variable at an output of an RF generator of thesystem of FIG. 1, in accordance with one embodiment described in thepresent disclosure.

FIG. 10 is a block diagram of a system in which the voltage and currentprobe and a communication device are located outside the RF generator,in accordance with an embodiment described in the present disclosure.

FIG. 11 is a block diagram of a system in which values of the variabledetermined using the system of FIG. 1 are used, in accordance with anembodiment described in the present disclosure.

FIG. 12A is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when an x MHz RF generator is on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 12B is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when a y MHz RF generator is on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 12C is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when a z MHz RF generator is on, in accordance with one embodimentdescribed in the present disclosure.

FIG. 13 is a flowchart of a method for determining wafer bias at a modelnode of the impedance matching model, the RF transmission model, or theESC model, in accordance with an embodiment described in the presentdisclosure.

FIG. 14 is a state diagram illustrating a wafer bias generator used togenerate a wafer bias, in accordance with an embodiment described in thepresent disclosure.

FIG. 15 is a flowchart of a method for determining a wafer bias at apoint along a path between the impedance matching model and the ESCmodel, in accordance with an embodiment described in the presentdisclosure.

FIG. 16 is a block diagram of a system for determining a wafer bias at anode of a model, in accordance with an embodiment described in thepresent disclosure.

FIG. 17 is a flowchart of a method for determining a wafer bias at amodel node of the system of FIG. 1, in accordance with an embodimentdescribed in the present disclosure.

FIG. 18 is a block diagram of a system that is used to illustrateadvantages of determining wafer bias by using the method of FIG. 13,FIG. 15, or FIG. 17 instead of by using a voltage probe, in accordancewith an embodiment described in the present disclosure.

FIG. 19A show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the yand z MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 19B show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the xand z MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 19C show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the xand y MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 20A is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model wafer bias thatis determined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x MHz RF generator is on, in accordance with anembodiment described in the present disclosure.

FIG. 20B is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the y MHz RF generator is on, in accordance with oneembodiment described in the present disclosure.

FIG. 20C is a diagram of embodiments of graphs used to illustrate acorrelation between a wired wafer bias measured using a sensor tool, amodel bias that is determined using the method of FIG. 13, 15, or 17 andan error in the model bias when the z MHz RF generator is on, inaccordance with one embodiment described in the present disclosure.

FIG. 20D is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x and y MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20E is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x and z MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20F is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the y and z MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20G is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x, y, and z MHz RF generators are on, in accordancewith an embodiment described in the present disclosure.

FIG. 21 is a block diagram of a host system of the system of FIG. 1, inaccordance with an embodiment described in the present disclosure.

FIG. 22 is a block diagram of the RF transmission system, in accordancewith an embodiment described in the present disclosure.

FIG. 23 is a simplified diagram of the electrostatic chuck, inaccordance with an embodiment described in the present disclosure.

FIG. 24 is a flowchart of the method operations for determining an RFtransmission line model for the electrostatic chuck, the plasma and theRF return path, in accordance with an embodiment described in thepresent disclosure.

FIG. 25 is a flowchart of the method operations for adding an end moduleto the baseline RF transmission line model, in accordance with anembodiment described in the present disclosure.

FIG. 26 is a flowchart of an alternate method operations for adding anend module to the baseline RF transmission line model, in accordancewith an embodiment described in the present disclosure.

FIG. 27 is a flowchart of the method operations for scoring each of therevised RF transmission line models, in accordance with an embodimentdescribed in the present disclosure.

DETAILED DESCRIPTION

The following embodiments describe systems and methods for using a modelto with a location of a fault within an RF transmission system in aplasma system. It will be apparent that the present embodiments may bepracticed without some or all of these specific details. In otherinstances, well known process operations have not been described indetail in order not to unnecessarily obscure the present embodiments.

FIG. 1 is a block diagram of an embodiment of a system 126 fordetermining a variable at an output of an impedance matching model 104,at an output, e.g., a model node N1 m, of a portion 173 of an RFtransmission model 161, which is a model of an RF transmission line 113,and at an output, e.g., a model node N6 m, of an electrostatic chuck(ESC) model 125. Examples of a variable include complex voltage, complexcurrent, complex voltage and current, complex power, wafer bias, etc.The RF transmission line 113 has an output, e.g., a node N2. A voltageand current (VI) probe 110 measures a complex voltage and current Vx,Ix, and φx, e.g., a first complex voltage and current, at an output,e.g., a node N3, of an x MHz RF generator. It should be noted that Vxrepresents a voltage magnitude, Ix represents a current magnitude, andφx represents a phase between Vx and Ix. The impedance matching model104 has an output, e.g., a model node N4 m.

Moreover, a voltage and current probe 111 measures a complex voltage andcurrent Vy, Iy, and φy at an output, e.g., a node N5, of a y MHz RFgenerator. It should be noted that Vy represents a voltage magnitude, Iyrepresents a current magnitude, and φy represents a phase between Vy andIy.

In some embodiments, a node is an input of a device, an output of adevice, or a point within the device. A device, as used herein, isdescribed below.

Examples of x MHz include 2 MHz, 27 MHz, and 60 MHz. Examples of y MHzinclude 2 MHz, 27 MHz, and 60 MHz. The x MHz is different than y MHz.For example, when x MHz is 2 MHz, y MHz is 27 MHz or 60 MHz. When x MHzis 27 MHz, y MHz is 60 MHz.

An example of each voltage and current probe 110 and 111 includes avoltage and current probe that complies with a pre-set formula. Anexample of the pre-set formula includes a standard that is followed byan Association, which develops standards for sensors. Another example ofthe pre-set formula includes a National Institute of Standards andTechnology (NIST) standard. As an illustration, the voltage and currentprobe 110 or 111 is calibrated according to NIST standard. In thisillustration, the voltage and current probe 110 or 111 is coupled withan open circuit, a short circuit, or a known load to calibrate thevoltage and current probe 110 or 111 to comply with the NIST standard.The voltage and current probe 110 or 111 may first be coupled with theopen circuit, then with the short circuit, and then with the known loadto calibrate the voltage and current probe 110 based on NIST standard.The voltage and current probe 110 or 111 may be coupled to the knownload, the open circuit, and the short circuit in any order to calibratethe voltage and current probe 110 or 111 according to NIST standard.Examples of a known load include a 50 ohm load, a 100 ohm load, a 200ohm load, a static load, a direct current (DC) load, a resistor, etc. Asan illustration, each voltage and current probe 110 and 111 iscalibrated according NIST-traceable standards.

The voltage and current probe 110 is coupled to the output, e.g., thenode N3, of the x MHz RF generator. The output, e.g., the node N3, ofthe x MHz RF generator is coupled to an input 153 of an impedancematching circuit 114 via a cable 150. Moreover, the voltage and currentprobe 111 is coupled to the output, e.g., the node N5, of the y MHz RFgenerator. The output, e.g., the node N5, of the y MHz RF generator iscoupled to another input 155 of the impedance matching circuit 114 via acable 152.

An output, e.g., a node N4, of the impedance matching circuit 114 iscoupled to an input of the RF transmission line 113. The RF transmissionline 113 includes a portion 169 and another portion 195. An input of theportion 169 is an input of the RF transmission line 113. An output,e.g., a node N1, of the portion 169 is coupled to an input of theportion 195. An output, e.g., the node N2, of the portion 195 is coupledto the plasma chamber 175. The output of the portion 195 is the outputof the RF transmission line 113. An example of the portion 169 includesan RF cylinder and an RF strap. The RF cylinder is coupled to the RFstrap. An example of the portion 195 includes an RF rod and/or asupport, e.g., a cylinder, etc., for supporting the plasma chamber 175.

The plasma chamber 175 includes an electrostatic chuck (ESC) 177, anupper electrode 179, and other parts (not shown), e.g., an upperdielectric ring surrounding the upper electrode 179, an upper electrodeextension surrounding the upper dielectric ring, a lower dielectric ringsurrounding a lower electrode of the ESC 177, a lower electrodeextension surrounding the lower dielectric ring, an upper plasmaexclusion zone (PEZ) ring, a lower PEZ ring, etc. The upper electrode179 is located opposite to and facing the ESC 177. A work piece 131,e.g., a semiconductor wafer, etc., is supported on an upper surface 183of the ESC 177. The upper surface 183 includes an output N6 of the ESC177. The work piece 131 is placed on the output N6. Various processes,e.g., chemical vapor deposition, cleaning, deposition, sputtering,etching, ion implantation, resist stripping, etc., are performed on thework piece 131 during production. Integrated circuits, e.g., applicationspecific integrated circuit (ASIC), programmable logic device (PLD),etc. are developed on the work piece 131 and the integrated circuits areused in a variety of electronic items, e.g., cell phones, tablets, smartphones, computers, laptops, networking equipment, etc. Each of the lowerelectrode and the upper electrode 179 is made of a metal, e.g.,aluminum, alloy of aluminum, copper, etc.

In one embodiment, the upper electrode 179 includes a hole that iscoupled to a central gas feed (not shown). The central gas feed receivesone or more process gases from a gas supply (not shown). Examples of aprocess gases include an oxygen-containing gas, such as O₂. Otherexamples of a process gas include a fluorine-containing gas, e.g.,tetrafluoromethane (CF₄), sulfur hexafluoride (SF₆), hexafluoroethane(C₂F₆), etc. The upper electrode 179 is grounded. The ESC 177 is coupledto the x MHz RF generator and the y MHz RF generator via the impedancematching circuit 114.

When the process gas is supplied between the upper electrode 179 and theESC 177 and when the x MHz RF generator and/or the y MHz RF generatorsupplies RF signals via the impedance matching circuit 114 and the RFtransmission line 113 to the ESC 177, the process gas is ignited togenerate plasma within the plasma chamber 175.

When the x MHz RF generator generates and provides an RF signal via thenode N3, the impedance matching circuit 114, and the RF transmissionline 113 to the ESC 177 and when the y MHz generator generates andprovides an RF signal via the node N5, the impedance matching circuit114, and the RF transmission line 113 to the ESC 177, the voltage andcurrent probe 110 measures the complex voltage and current at the nodeN3 and the voltage and current probe 111 measures the complex voltageand current at the node N5.

The complex voltages and currents measured by the voltage and currentprobes 110 and 111 are provided via corresponding communication devices185 and 189 from the corresponding voltage and current probes 110 and111 to a storage hardware unit (HU) 162 of a host system 130 forstorage. For example, the complex voltage and current measured by thevoltage and current probe 110 is provided via the communication device185 and a cable 191 to the host system 130 and the complex voltage andcurrent measured by the voltage and current probe 111 is provided viathe communication device 189 and a cable 193 to the host system 130.Examples of a communication device include an Ethernet device thatconverts data into Ethernet packets and converts Ethernet packets intodata, an Ethernet for Control Automation Technology (EtherCAT) device, aserial interface device that transfers data in series, a parallelinterface device that transfers data in parallel, a Universal Serial Bus(USB) interface device, etc.

Examples of the host system 130 include a computer, e.g., a desktop, alaptop, a tablet, etc. As an illustration, the host system 130 includesa processor and the storage HU 162. As used herein, a processor may be acentral processing unit (CPU), a microprocessor, an application specificintegrated circuit (ASIC), a programmable logic device (PLD), etc.Examples of the storage HU include a read-only memory (ROM), a randomaccess memory (RAM), or a combination thereof. The storage HU may be aflash memory, a redundant array of storage disks (RAID), a hard disk,etc.

The impedance matching model 104 is stored within the storage HU 162.The impedance matching model 104 has similar characteristics, e.g.,capacitances, inductances, complex power, complex voltage and currents,etc., as that of the impedance matching circuit 114. For example, theimpedance matching model 104 has the same number of capacitors and/orinductors as that within the impedance matching circuit 114, and thecapacitors and/or inductors are connected with each other in the samemanner, e.g., serial, parallel, etc. as that within the impedancematching circuit 114. To provide an illustration, when the impedancematching circuit 114 includes a capacitor coupled in series with aninductor, the impedance matching model 104 also includes the capacitorcoupled in series with the inductor.

As an example, the impedance matching circuit 114 includes one or moreelectrical components and the impedance matching model 104 includes adesign, e.g., a computer-generated model, of the impedance matchingcircuit 114. The computer-generated model may be generated by aprocessor based upon input signals received from a user via an inputhardware unit. The input signals include signals regarding whichelectrical components, e.g., capacitors, inductors, etc., to include ina model and a manner, e.g., series, parallel, etc., of coupling theelectrical components with each other. As another example, the impedancecircuit 114 includes hardware electrical components and hardwareconnections between the electrical components and the impedance matchingmodel 104 include software representations of the hardware electricalcomponents and of the hardware connections. As yet another example, theimpedance matching model 104 is designed using a software program andthe impedance matching circuit 114 is made on a printed circuit board.As used herein, electrical components may include resistors, capacitors,inductors, connections between the resistors, connections between theinductors, connections between the capacitors, and/or connectionsbetween a combination of the resistors, inductors, and capacitors.

Similarly, a cable model 163 and the cable 150 have similarcharacteristics, and a cable model 165 and the cable 152 has similarcharacteristics. As an example, an inductance of the cable model 163 isthe same as an inductance of the cable 150. As another example, thecable model 163 is a computer-generated model of the cable 150 and thecable model 165 is a computer-generated model of the cable 152.

Similarly, an RF transmission model 161 and the RF transmission line 113have similar characteristics. For example, the RF transmission model 161has the same number of resistors, capacitors and/or inductors as thatwithin the RF transmission line 113, and the resistors, capacitorsand/or inductors are connected with each other in the same manner, e.g.,serial, parallel, etc. as that within the RF transmission line 113. Tofurther illustrate, when the RF transmission line 113 includes acapacitor coupled in parallel with an inductor, the RF transmissionmodel 161 also includes the capacitor coupled in parallel with theinductor. As yet another example, the RF transmission line 113 includesone or more electrical components and the RF transmission model 161includes a design, e.g., a computer-generated model, of the RFtransmission line 113.

In some embodiments, the RF transmission model 161 is acomputer-generated impedance transformation involving computation ofcharacteristics, e.g., capacitances, resistances, inductances, acombination thereof, etc., of elements, e.g., capacitors, inductors,resistors, a combination thereof, etc., and determination ofconnections, e.g., series, parallel, etc., between the elements.

Based on the complex voltage and current received from the voltage andcurrent probe 110 via the cable 191 and characteristics, e.g.,capacitances, inductances, etc., of elements, e.g., inductors,capacitors, etc., within the impedance matching model 104, the processorof the host system 130 calculates a complex voltage and current V, I,and φ, e.g., a second complex voltage and current, at the output, e.g.,the model node N4 m, of the impedance matching model 104. The complexvoltage and current at the model node N4 m is stored in the storage HU162 and/or another storage HU, e.g., a compact disc, a flash memory,etc., of the host system 130. The complex V, I, and φ includes a voltagemagnitude V, a current magnitude I, and a phase φ between the voltageand current.

The output of the impedance matching model 104 is coupled to an input ofthe RF transmission model 161, which is stored in the storage hardwareunit 162. The impedance matching model 104 also has an input, e.g., anode N3 m, which is used to receive the complex voltage and currentmeasured at the node N3.

The RF transmission model 161 includes the portion 173, another portion197, and an output N2 m, which is coupled via the ESC model 125 to themodel node N6 m. The ESC model 125 is a model of the ESC 177. Forexample, the ESC model 125 has similar characteristics as that of theESC 177. For example, the ESC model 125 has the same inductance,capacitance, resistance, or a combination thereof as that of the ESC177.

An input of the portion 173 is the input of the RF transmission model161. An output of the portion 173 is coupled to an input of the portion197. The portion 172 has similar characteristics as that of the portion169 and the portion 197 has similar characteristics as that of theportion 195.

Based on the complex voltage and current measured at the model node N4m, the processor of the host system 130 calculates a complex voltage andcurrent V, I, and φ, e.g., a third complex voltage and current, at theoutput, e.g., the model node N1 m, of the portion 173 of the RFtransmission model 161. The complex voltage and current determined atthe model node N1 m is stored in the storage HU 162 and/or anotherstorage HU, e.g., a compact disc, a flash memory, etc., of the hostsystem 130.

In several embodiments, instead of or in addition to determining thethird complex voltage and current, the processor of the host system 130computes a complex voltage and current, e.g., an intermediate complexvoltage and current V, I, and φ, at a point, e.g., a node, etc., withinthe portion 173 based on the complex voltage and current at the outputof the impedance matching model 104 and characteristics of elementsbetween the input of the RF transmission model 161 and the point withinthe portion 173.

In various embodiments, instead of or in addition to determining thethird complex voltage and current, the processor of the host system 130computes a complex voltage and current, e.g., an intermediate complexvoltage and current V, I, and φ, at a point, e.g., a node, etc., withinthe portion 197 based on the complex voltage and current at the outputof the impedance matching model 104 and characteristics of elementsbetween the input of the RF transmission model 161 and the point withinthe portion 197.

It should be noted that in some embodiments, the complex voltage andcurrent at the output of the impedance matching model 104 is calculatedbased on the complex voltage and current at the output of the x MHz RFgenerator, characteristics of elements the cable model 163, andcharacteristics of the impedance matching model 104.

It should further be noted that although two generators are showncoupled to the impedance matching circuit 114, in one embodiment, anynumber of RF generators, e.g., a single generator, three generators,etc., are coupled to the plasma chamber 175 via an impedance matchingcircuit. For example, a 2 MHz generator, a 27 MHz generator, and a 60MHz generator may be coupled to the plasma chamber 175 via an impedancematching circuit. For example, although the above-described embodimentsare described with respect to using complex voltage and current measuredat the node N3, in various embodiments, the above-described embodimentsmay also use the complex voltage and current measured at the node N5.

FIG. 2 is a flowchart of an embodiment of a method 102 for determiningthe complex voltage and current at the output of the RF transmissionmodel portion 173 (FIG. 1). The method 102 is executed by the processorof the host system 130 (FIG. 1). In an operation 106, the complexvoltage and current, e.g., the first complex voltage and current,measured at the node N3 is identified from within the storage HU 162(FIG. 1). For example, it is determined that the first complex voltageand current is received from the voltage and current probe 110 (FIG. 1).As another example, based on an identity, of the voltage and currentprobe 110, stored within the storage HU 162 (FIG. 1), it is determinedthat the first complex voltage and current is associated with theidentity.

Furthermore, in an operation 107, the impedance matching model 104(FIG. 1) is generated based on electrical components of the impedancematching circuit 114 (FIG. 1). For example, connections betweenelectrical components of the impedance matching circuit 114 andcharacteristics of the electrical components are provided to theprocessor of the host system 130 by the user via an input hardware unitthat is coupled with the host system 130. Upon receiving the connectionsand the characteristics, the processor generates elements that have thesame characteristics as that of electrical components of the impedancematching circuit 114 and generates connections between the elements thathave the same connections as that between the electrical components.

The input, e.g., the node N3 m, of the impedance matching model 163receives the first complex voltage and current. For example, theprocessor of the host system 130 accesses, e.g., reads, etc., from thestorage HU 162 the first complex voltage and current and provides thefirst complex voltage and current to the input of the impedance matchingmodel 104 to process the first complex voltage and current.

In an operation 116, the first complex voltage and current is propagatedthrough one or more elements of the impedance matching model 104(FIG. 1) from the input, e.g., the node N3 m (FIG. 1), of the impedancematching model 104 to the output, e.g., the node N4 m (FIG. 1), of theimpedance matching model 104 to determine the second complex voltage andcurrent, which is at the output of the impedance matching model 104. Forexample, with reference to FIG. 3B, when the 2 MHz RF generator is on,e.g., operational, powered on, coupled to the devices, such as, forexample, the impedance matching circuit 104, of the plasma system 126,etc., a complex voltage and current Vx1, Ix1, and φx1, e.g., anintermediate complex voltage and current, which includes the voltagemagnitude Vx1, the current magnitude Ix1, and the phase φx1 between thecomplex voltage and current, at a node 251, e.g., an intermediate node,is determined based on a capacitance of a capacitor 253, based on acapacitance of a capacitor C5, and based on the first complex voltageand current that is received at an input 255. Moreover, a complexvoltage and current Vx2, Ix2, and φx2 at a node 257 is determined basedon the complex voltage and current Vx1, Ix1, and φx1, and based on aninductance of an inductor L3. The complex voltage and current Vx2, Ix2,and φx2 includes the voltage magnitude Vx2, the current magnitude Ix2,and the phase φx2 between the voltage and current. When the 27 MHz RFgenerator and the 60 MHz RF generator are off, e.g., nonoperational,powered off, decoupled from the impedance matching circuit 104, etc., acomplex voltage and current V2, I2, and φ2 is determined to be thesecond complex voltage and current at an output 259, which is an exampleof the output, e.g., the model node N4 m (FIG. 1), of the impedancematching model 104 (FIG. 1). The complex voltage and current V2, I2, andφ2 is determined based on the complex voltage and current Vx2, Ix2, andφx2 and an inductor of an inductor L2. The complex voltage and currentV2, I2, and φ2 includes the voltage magnitude V2, the current magnitudeI2, and the phase φ2 between the voltage and current.

Similarly, when 27 MHz RF generator is on and the 2 MHz and the 60 MHzRF generators are off, a complex voltage and current V27, I27, and φ27at the output 259 is determined based on a complex voltage and currentreceived at a node 261 and characteristics of an inductor LPF2, acapacitor C3, a capacitor C4, and an inductor L2. The complex voltageand current V27, I27, and φ27 includes the voltage magnitude V27, thecurrent magnitude I27, and the phase φ27 between the voltage andcurrent. The complex voltage and current received at the node 261 is thesame as the complex voltage and current measured at the node N5 (FIG.1). When both the 2 MHz and 27 MHz RF generators are on and the 60 MHzRF generator is off, the complex voltages and currents V2, I2, φ2, V27,I27, and φ27 are an example of the second complex voltage and current.Moreover, similarly, when the 60 MHz RF generator is on and the 2 and 27MHz RF generators are off, a complex voltage and current V60, I60, andφ60 at the output 259 is determined based on a complex voltage andcurrent received at a node 265 and characteristics of an inductor LPF1,a capacitor C1, a capacitor C2, an inductor L4, a capacitor 269, and aninductor L1. The complex voltage and current V60, I60, and φ60 includesthe voltage magnitude V60, the current magnitude I60, and the phase φ60between the voltage and current. When the 2 MHz, 27 MHz, and the 60 MHzRF generators are on, the complex voltages and currents V2, I2, φ2, V27,I27, φ27, V60, I60, and φ60 are an example of the second complex voltageand current.

In an operation 117, the RF transmission model 161 (FIG. 1) is generatedbased on the electrical components of the RF transmission line 113 (FIG.1). For example, connections between electrical components of the RFtransmission line 113 and characteristics of the electrical componentsare provided to the processor of the host system 130 by the user via aninput device that is coupled with the host system 130. Upon receivingthe connections and the characteristics, the processor generateselements that have the same characteristics as that of electricalcomponents of the RF transmission line 113 and generates connectionsbetween the elements that are the same as that between the electricalcomponents.

In an operation 119, the second complex voltage and current ispropagated through one or more elements of the RF transmission modelportion 173 from the input of the RF transmission model 113 to theoutput, e.g., the model node N1 m (FIG. 1), of the RF transmission modelportion 173 to determine the third complex voltage and current at theoutput of the RF transmission model portion 173. For example, withreference to FIG. 5B, when the 2 MHz RF generator is on and the 27 and60 MHz RF generators are off, a complex voltage and current Vx4, Ix4,and φx4, e.g., an intermediate complex voltage and current, at a node293, e.g., an intermediate node, is determined based on an inductance ofan inductor Ltunnel, based on a capacitance of a capacitor Ctunnel, andbased on the complex voltage and current V2, I2, and φ2 (FIG. 3B), whichis an example of the second complex voltage and current. It should benoted that Ltunnel is an inductance of a computer-generated model of anRF tunnel and Ctunnel is a capacitance of the RF tunnel model. Moreover,a complex voltage and current V21, I21, and φ21 at an output 297 of atunnel and strap model 210 is determined based on the complex voltageand current Vx4, Ix4, and φx4, and based on an inductance of an inductorLstrap. The output 297 is an example of the output, e.g., the model nodeN1 m (FIG. 1), of the portion 173 (FIG. 1). It should be noted thatLstrap is an inductance of a computer-generated model of the RF strap.When the 2 MHz RF generator is on and the 27 and 60 MHz RF generatorsare off, the complex voltage and current V21, I21, and φ21 is determinedto be the third complex voltage and current at the output 297.

Similarly, when the 27 MHz RF generator is on and the 2 and 60 MHz RFgenerators are off, a complex voltage and current V271, I271, and φ271at the output 297 is determined based on the complex voltage and currentV27, I27, φ27 (FIG. 3B) at the output 259 and characteristics of theinductor Ltunnel, the capacitor Ctunnel, and the inductor Lstrap. Whenboth the 2 MHz and 27 MHz RF generators are on and the 60 MHz RFgenerator is off, the complex voltages and currents V21, I21, φ21, V271,I271, and φ271 are an example of the third complex voltage and current.

Moreover, similarly, when the 60 MHz RF generator is powered on and the2 and 27 MHz RF generators are powered off, a complex voltage andcurrent V601, I601, and φ601 at the output 297 is determined based onthe complex voltage and current V60, I60, and φ60 (FIG. 3B) received ata node 259 and characteristics of the inductor Ltunnel, the capacitorCtunnel, and the inductor Lstrap. When the 2 MHz, 27 MHz, and the 60 MHzRF generators are on, the complex voltages and currents V21, I21, φ21,V271, I271, φ71, V601, I601, and φ601 are an example of the thirdcomplex voltage and current. The method 102 ends after the operation119.

FIG. 3A is a block diagram of an embodiment of a system 123 used toillustrate an impedance matching circuit 122. The impedance matchingcircuit 122 is an example of the impedance matching circuit 114 (FIG.1). The impedance matching circuit 122 includes series connectionsbetween electrical components and/or parallel connections betweenelectrical components.

FIG. 3B is a circuit diagram of an embodiment of an impedance matchingmodel 172. The impedance matching model 172 is an example of theimpedance matching model 104 (FIG. 1). As shown, the impedance matchingmodel 172 includes capacitors having capacitances C1 thru C9, inductorshaving inductances LPF1, LPF2, and L1 thru L4. It should be noted thatthe manner in which the inductors and/or capacitors are coupled witheach other in FIG. 3B is an example. For example, the inductors and/orcapacitors shown in FIG. 3B can be coupled in a series and/or parallelmanner with each other. Also, in some embodiments, the impedancematching model 172 includes a different number of capacitors and/or adifferent number of inductors than that shown in FIG. 3B.

FIG. 4 is a diagram of an embodiment of a system 178 used to illustratean RF transmission line 181, which is an example of the RF transmissionline 113 (FIG. 1). The RF transmission line 181 includes a cylinder 148,e.g., a tunnel. Within a hollow of the cylinder 148 lies an insulator189 and an RF rod 142. A combination of the cylinder 148 and the RF rod142 is an example of the portion 169 (FIG. 1) of the RF transmissionline 113 (FIG. 1). The RF transmission line 181 is bolted via bolts B1,B2, B3, and B4 with the impedance matching circuit 114. In oneembodiment, the RF transmission line 181 is bolted via any number ofbolts with the impedance matching circuit 114. In some embodiments,instead of or in addition to bolts, any other form of attachment, e.g.,glue, screws, etc., is used to attach the RF transmission line 181 tothe impedance matching circuit 114.

The RF transmission rod 142 is coupled with the output of the impedancematching circuit 114. Also, an RF strap 144, also known as RF spoon, iscoupled with the RF rod 142 and with an RF rod 199, a portion of whichis located within a support 146, e.g., a cylinder. The support 146 thatincludes the RF rod 199 is an example of the portion 195 (FIG. 1). In anembodiment, a combination of the cylinder 148, the RF rod 142, the RFstrap 144, the support 146 and the RF rod 199 forms the RF transmissionline 181, which is an example of the RF transmission line 113 (FIG. 1).The support 146 provides support to the plasma chamber. The support 146is attached to the ESC 177 of the plasma chamber. An RF signal issupplied from the x MHz generator via the cable 150, the impedancematching circuit 114, the RF rod 142, the RF strap 144, and the RF rod199 to the ESC 177.

In one embodiment, the ESC 177 includes a heating element and anelectrode on top of the heating element. In an embodiment, the ESC 177includes a heating element and the lower electrode. In one embodiment,the ESC 177 includes the lower electrode and a heating element, e.g.,coil wire, etc., embedded within holes formed within the lowerelectrode. In some embodiments, the electrode is made of a metal, e.g.,aluminum, copper, etc. It should be noted that the RF transmission line181 supplies an RF signal to the lower electrode of the ESC 177.

FIG. 5A is a block diagram of an embodiment of a system 171 used toillustrate a circuit model 176 of the RF transmission line 113 (FIG. 1).For example, the circuit model 176 includes inductors and/or capacitors,connections between the inductors, connections between the capacitors,and/or connections between the inductors and the capacitors. Examples ofconnections include series and/or parallel connections. The circuitmodel 176 is an example of the RF transmission model 161 (FIG. 1).

FIG. 5B is a diagram of an embodiment of an electrical circuit 180 usedto illustrate the tunnel and strap model 210, which is an example of theportion 173 (FIG. 1) of the RF transmission model 161 (FIG. 1). Theelectrical circuit 180 includes the impedance matching model 172 and thetunnel and strap model 210. The tunnel and strap model 210 includesinductors Ltunnel and Lstrap and a capacitor Ctunnel. It should be notedthat the inductor Ltunnel represents an inductance of the cylinder 148(FIG. 4) and the RF rod 142 and the capacitor Ctunnel represents acapacitance of the cylinder 148 and the RF rod 142. Moreover, theinductor Lstrap represents an inductance of the RF strap 144 (FIG. 4).

In an embodiment, the tunnel and strap model 210 includes any number ofinductors and/or any number of capacitors. In this embodiment, thetunnel and strap model 210 includes any manner, e.g., serial, parallel,etc. of coupling a capacitor to another capacitor, coupling a capacitorto an inductor, and/or coupling an inductor to another inductor.

FIG. 5C is a diagram of an embodiment of an electrical circuit 300 usedto illustrate a tunnel and strap model 302, which is an example of theportion 173 (FIG. 1) of the RF transmission model 161 (FIG. 1). Thetunnel and strap model 302 is coupled via the output 259 to theimpedance matching model 172. The tunnel and strap model 302 includesinductors having inductances 20 nanoHenry (nH) and capacitors havingcapacitances of 15 picoFarads (pF), 31 pF, 15.5 pF, and 18.5 pF. Thetunnel and strap model 302 is coupled via a node 304 to an RF cylinder,which is coupled to the ESC 177 (FIG. 1). The RF cylinder is an exampleof the portion 195 (FIG. 1).

It should be noted that in some embodiments, the inductors andcapacitors of the tunnel and strap model 302 have other values. Forexample, the 20 nH inductors have an inductance ranging between 15 and20 nH or between 20 and 25 nH. As another example, two or more of theinductors of the tunnel and strap model 302 have difference inductances.As yet another example, the 15 pF capacitor has a capacitance rangingbetween 8 pF and 25 pF, the 31 pF capacitor has a capacitance rangingbetween 15 pF and 45 pF, the 15.5 pF capacitor has a capacitance rangingbetween 9 pF and 20 pF, and the 18.5 pF capacitor has a capacitanceranging between 10 pF and 27 pF.

In various embodiments, any number of inductors are included in thetunnel and strap model 302 and any number of capacitors are included inthe tunnel and strap model 302.

FIG. 6 is a diagram of an embodiment of an electrical circuit 310 usedto illustrate a cylinder and ESC model 312, which is a combination of aninductor 313 and a capacitor 316. The cylinder and ESC model 312includes a cylinder model and an ESC model, which is an example of theESC model 125 (FIG. 1). The cylinder model is an example of the portion197 (FIG. 1) of the RF transmission model 161 (FIG. 1). The cylinder andESC model 312 has similar characteristics as that of a combination ofthe portion 195 and the ESC 177 (FIG. 1). For example, the cylinder andESC model 312 has the same resistance as that of a combination of theportion 195 and the ESC 177. As another example, the cylinder and ESCmodel 312 has the same inductance as that of a combination of theportion 195 and the ESC 177. As yet another example, the cylinder andESC model 312 has the same capacitance as that of a combination of theportion 195 and the ESC 177. As yet another example, the cylinder andESC model 312 has the same inductance, resistance, capacitance, or acombination thereof, as that of a combination of the portion 195 and theESC 177.

The cylinder and ESC model 312 is coupled via a node 318 to the tunneland strap model 302. The node 318 is an example of the model node N1 m(FIG. 1).

It should be noted that in some embodiments, an inductor having aninductance other than the 44 milliHenry (mH) is used in the cylinder andESC model 312. For example, an inductor having an inductance rangingfrom 35 mH to 43.9 mH or from 45.1 mH too 55 mH is used. In variousembodiments, a capacitor having a capacitance other than 550 pF is used.For example, instead of the 550 pF capacitor, a capacitor having acapacitance ranging between 250 and 550 pF or between 550 and 600 pF isused.

The processor of the host system 130 (FIG. 1) calculates a combinedimpedance, e.g., total impedance, etc., of a combination of the model172, the tunnel and strap model 302, and the cylinder and ESC model 312.The combined impedance and complex voltage and current determined at themodel node 318 are used as inputs by the processor of the host system130 to calculate a complex voltage and impedance at the node N6 m. Itshould be noted that an output of the cylinder and ESC model 312 is themodel node N6 m.

FIG. 7 is a block diagram of an embodiment of a system 200 that is usedto determine a variable. The system 200 includes a plasma chamber 135,which further includes an ESC 201 and has an input 285. The plasmachamber 135 is an example of the plasma chamber 175 (FIG. 1) and the ESC201 is an example of the ESC 177 (FIG. 1). The ESC 201 includes aheating element 198. Also, the ESC 201 is surrounded by an edge ring(ER) 194. The ER 194 includes a heating element 196. In an embodiment,the ER 194 facilitates a uniform etch rate and reduced etch rate driftnear an edge of the workpiece 131 that is supported by the ESC 201.

A power supply 206 provides power to the heating element 196 via afilter 208 to heat the heating element 196 and a power supply 204provides power to the heating element 198 via a filter 202 to heat theheating element 198. In an embodiment, a single power supply providespower to both the heating elements 196 and 198. The filter 208 filtersout predetermined frequencies of a power signal that is received fromthe power supply 206 and the filter 202 filters out predeterminedfrequencies of a power signal that is received from the power supply204.

The heating element 198 is heated by the power signal received from thepower supply 204 to maintain an electrode of the ESC 198 at a desirabletemperature to further maintain an environment within the plasma chamber135 at a desirable temperature. Moreover, the heating element 196 isheated by the power signal received from the power supply 206 tomaintain the ER 194 at a desirable temperature to further maintain anenvironment within the plasma chamber 135 at a desirable temperature.

It should be noted that in an embodiment, the ER 194 and the ESC 201include any number of heating elements and any type of heating elements.For example, the ESC 201 includes an inductive heating element or ametal plate. In one embodiment, each of the ESC 201 and the ER 194includes one or more cooling elements, e.g., one or more tubes thatallow passage of cold water, etc., to maintain the plasma chamber 135 ata desirable temperature.

It should further be noted that in one embodiment, the system 200includes any number of filters. For example, the power supplies 204 and206 are coupled to the ESC 201 and the ER 194 via a single filter.

FIG. 8A is a diagram of an embodiment of a system 217 used to illustratea model of the filters 202 and 208 (FIG. 7) to improve an accuracy ofthe variable. The system 217 includes the tunnel and strap model 210that is coupled via a cylinder model 211 to a model 216, which includescapacitors and/or inductors and connections therebetween of the filters202 and 208. The model 216 is stored within the storage HU 162 (FIG. 1)and/or the other storage HU. The capacitors and/or inductors of themodel 216 are coupled with each other in a manner, e.g., a parallelmanner, a serial manner, a combination thereof, etc. The model 216represents capacitances and/or inductances of the filters 202 and 208.

Moreover, the system 217 includes the cylinder model 211, which is acomputer-generated model of the RF rod 199 (FIG. 4) and the support 146(FIG. 4). The cylinder model 211 has similar characteristics as that ofelectrical components of the RF rod 199 and the support 146. Thecylinder model 211 includes one or more capacitors, one or moreinductors, connections between the inductors, connections between thecapacitors, and/or connections between a combination of the capacitorsand inductors.

The processor of the host system 130 (FIG. 1) calculates a combinedimpedance, e.g., total impedance, etc., of the model 216, the tunnel andstrap model 210, and the cylinder model 211. The combined impedanceprovides a complex voltage and impedance at the node N2 m. With theinclusion of the model 216 and the tunnel and strap model 214 indetermining the variable at the node N2 m, accuracy of the variable isimproved. It should be noted that an output of the model 216 is themodel node N2 m.

FIG. 8B is a diagram of an embodiment of a system 219 used to illustratea model of the filters 202 and 208 (FIG. 7) to improve an accuracy ofthe variable. The system 219 includes the tunnel and strap model 210 anda model 218, which is coupled in parallel to the tunnel and strap model210. The model 218 is an example of the model 216 (FIG. 8A). The model218 includes an inductor Lfilter, which represents a combined inductanceof the filters 202 and 208. The model 218 further includes a capacitorCfilter, which represents directed combined capacitance of the filters202 and 208.

FIG. 9 is a block diagram of an embodiment of a system 236 for using avoltage and current probe 238 to measure a variable at an output 231 ofan RF generator 220. The output 231 is an example of the node N3(FIG. 1) or of the node N5 (FIG. 1). The RF generator 220 is an exampleof the x MHz generator or the y MHz generator (FIG. 1). The host system130 generates and provides a digital pulsing signal 213 having two ormore states to a digital signal processor (DSP) 226. In one embodiment,the digital pulsing signal 213 is a transistor-transistor logic (TTL)signal. Examples of the states include an on state and an off state, astate having a digital value of 1 and a state having a digital value of0, a high state and a low state, etc.

In another embodiment, instead of the host system 130, a clockoscillator, e.g., a crystal oscillator, etc., is used to generate ananalog clock signal, which is converted by an analog-to-digitalconverter into a digital signal similar to the digital pulsing signal213.

The digital pulsing signal 213 is sent to the DSP 226. The DSP 226receives the digital pulsing signal 213 and identifies the states of thedigital pulsing signal 213. For example, the DSP 226 determines that thedigital pulsing signal 213 has a first magnitude, e.g., the value of 1,the high state magnitude, etc., during a first set of time periods andhas a second magnitude, e.g., the value of 0, the low state magnitude,etc., during a second set of time periods. The DSP 226 determines thatthe digital pulsing signal 213 has a state S1 during the first set oftime periods and has a state S0 during the second set of time periods.Examples of the state S0 include the low state, the state having thevalue of 0, and the off state. Examples of the state S1 include the highstate, the state having the value of 1, and the on state. As yet anotherexample, the DSP 226 compares a magnitude of the digital pulsing signal213 with a pre-stored value to determine that the magnitude of thedigital pulsing signal 213 is greater than the pre-stored value duringthe first set of time periods and that the magnitude during the state S0of the digital pulsing signal 213 is not greater than the pre-storedvalue during the second set of time periods. In the embodiment in whichthe clock oscillator is used, the DSP 226 receives an analog clocksignal from the clock oscillator, converts the analog signal into adigital form, and then identifies the two states S0 and S1.

When a state is identified as S1, the DSP 226 provides a power value P1and/or a frequency value F1 to a parameter control 222. Moreover, whenthe state is identified as S0, the DSP 226 provides a power value P0and/or a frequency value F0 to a parameter control 224. An example of aparameter control that is used to tune a frequency includes an autofrequency tuner (AFT).

It should be noted that the parameter control 222, the parameter control224, and the DSP 226 are portions of a control system 187. For example,the parameter control 222 and the parameter control 224 are logicblocks, e.g., tuning loops, etc., which are portions of a computerprogram that is executed by the DSP 226. In some embodiments, thecomputer program is embodied within a non-transitory computer-readablemedium, e.g., a storage HU.

In an embodiment, a controller, e.g., hardware controller, ASIC, PLD,etc., is used instead of a parameter control. For example, a hardwarecontroller is used instead of the parameter control 222 and anotherhardware controller is used instead of the parameter control 224.

Upon receiving the power value P1 and/or the frequency value F1, theparameter control 222 provides the power value P1 and/or the frequencyvalue F1 to a driver 228 of a drive and amplifier system (DAS) 232.Examples of a driver includes a power driver, a current driver, avoltage driver, a transistor, etc. The driver 228 generates an RF signalhaving the power value P1 and/or the frequency value F1 and provides theRF signal to an amplifier 230 of the DAS 232.

In one embodiment, the driver 228 generates an RF signal having a drivepower value that is a function of the power value P1 and/or having adrive frequency value that is a function of the frequency value F1. Forexample, the drive power value is within a few watts, e.g. 1 thru 5watts, etc., of the power value P1 and the drive frequency value iswithin a few Hz, e.g. 1 thru 5 Hz, etc., of the frequency value F1.

The amplifier 230 amplifies the RF signal having the power value P1and/or the frequency value F1 and generates an RF signal 215 thatcorresponds to the RF signal received from the driver 228. For example,the RF signal 215 has a higher amount of power than that of the powervalue P1. As another example, the RF signal 215 has the same amount ofpower as that of the power value P1. The RF signal 215 is transferredvia a cable 217 and the impedance matching circuit 114 to the ESC 177(FIG. 1).

The cable 217 is an example of the cable 150 or the cable 152 (FIG. 1).For example, when the RF generator 220 is an example of the x MHz RFgenerator (FIG. 1), the cable 217 is an example of the cable 150 andwhen the RF generator 220 is an example of the y MHz RF generator (FIG.1), the cable 217 is an example of the cable 152.

When the power value P1 and/or the frequency value F1 are provided tothe DAS 232 by the parameter control 222 and the RF signal 215 isgenerated, the voltage and current probe 238 measures values of thevariable at the output 231 that is coupled to the cable 217. The voltageand current probe 238 is an example of the voltage and current probe 110or the voltage and current probe 111 (FIG. 1). The voltage and currentprobe 238 sends the values of the variable via a communication device233 to the host system 130 for the host system 130 to execute the method102 (FIG. 3) and methods 340, 351, and 363 (FIGS. 13, 15, and 17)described herein. The communication device 233 is an example of thecommunication device 185 or 189 (FIG. 1). The communication device 233applies a protocol, e.g., Ethernet, EtherCAT, USB, serial, parallel,packetization, depacketization, etc., to transfer data from the voltageand current probe 238 to the host system 130. In various embodiments,the host system 130 includes a communication device that applies theprotocol applied by the communication device 233. For example, when thecommunication 233 applies packetization, the communication device of thehost system 130 applies depacketization. As another example, when thecommunication device 233 applies a serial transfer protocol, thecommunication device of the host system 130 applies a serial transferprotocol.

Similarly, upon receiving the power value P0 and/or the frequency valueF0, the parameter control 224 provides the power value P0 and/or thefrequency value F0 to the driver 228. The driver 228 creates an RFsignal having the power value P0 and/or the frequency value F0 andprovides the RF signal to the amplifier 230.

In one embodiment, the driver 228 generates an RF signal having a drivepower value that is a function of the power value P0 and/or having adrive frequency value that is a function of the frequency value F0. Forexample, the drive power value is within a few, e.g. 1 thru 5, watts ofthe power value P0 and the drive frequency value is within a few, e.g. 1thru 5, Hz of the frequency value F0.

The amplifier 230 amplifies the RF signal having the power value P0and/or the frequency value F0 and generates an RF signal 221 thatcorresponds to the RF signal received from the driver 228. For example,the RF signal 221 has a higher amount of power than that of the powervalue P0. As another example, the RF signal 221 has the same amount ofpower as that of the power value P0. The RF signal 221 is transferredvia the cable 217 and the impedance matching circuit 114 to the knownload 112 (FIG. 2).

When the power value P0 and/or the frequency value F0 are provided tothe DAS 232 by the parameter control 222 and the RF signal 121 isgenerated, the voltage and current probe 238 measures values of thevariable at the output 231. The voltage and current probe 238 sends thevalues of the variable to the host system 130 for the host system 130 toexecute the method 102 (FIG. 2), the method 340 (FIG. 13), the method351 (FIG. 15), or the method 363 (FIG. 17).

It should be noted that the in one embodiment, the voltage and currentprobe 238 is decoupled from the DSP 226. In some embodiments, thevoltage and current probe 238 is coupled to the DSP 226. It shouldfurther be noted that the RF signal 215 generated during the state S1and the RF signal 221 generated during the state S0 are portions of acombined RF signal. For example, the RF signal 215 is a portion of thecombined RF signal that has a higher amount of power than the RF signal221, which is another portion of the combined RF signal.

FIG. 10 is a block diagram of an embodiment of a system 250 in which thevoltage and current probe 238 and the communication device 233 arelocated outside the RF generator 220. In FIG. 1, the voltage and currentprobe 110 is located within the x MHz RF generator to measure thevariable at the output of the x MHz RF generator. The voltage andcurrent probe 238 is located outside the RF generator 220 to measure thevariable at the output 231 of the RF generator 220. The voltage andcurrent probe 238 is associated, e.g., coupled, to the output 231 of theRF generator 220.

FIG. 11 is a block diagram of an embodiment of a system 128 in which thevalues of the variable determined using the system 126 of FIG. 1 areused. The system 128 includes an m MHz RF generator, an n MHz RFgenerator, an impedance matching circuit 115, an RF transmission line287, and a plasma chamber 134. The plasma chamber 134 may be similar tothe plasma chamber 175.

It should be noted that in an embodiment, the x MHz RF generator of FIG.2 is similar to the m MHz RF generator and the y MHz RF generator ofFIG. 2 is similar to the n MHz RF generator. As an example, x MHz isequal to m MHz and y MHz is equal to n MHz. As another example, the xMHz generator and the m MHz generators have similar frequencies and they MHz generator and the n MHz generator have similar frequencies. Anexample of similar frequencies is when the x MHz is within a window,e.g., within kHz or Hz, of the m MHz frequency. In some embodiments, thex MHz RF generator of FIG. 2 is not similar to the m MHz RF generatorand the y MHz RF generator of FIG. 2 is not similar to the n MHz RFgenerator.

It is further noted that in various embodiments, a different type ofsensor is used in each of the m MHz and n MHz RF generators than thatused in each of the x MHz and y MHz RF generators. For example, a sensorthat does not comply with the NIST standard is used in the m MHz RFgenerator. As another example, a voltage sensor that measures onlyvoltage is used in the m MHz RF generator.

It should further be noted that in an embodiment, the impedance matchingcircuit 115 is similar to the impedance matching circuit 114 (FIG. 1).For example, an impedance of the impedance matching circuit 114 is thesame as an impedance of the impedance matching circuit 115. As anotherexample, an impedance of the impedance matching circuit 115 is within awindow, e.g., within 10-20%, of the impedance of the impedance matchingcircuit 114. In some embodiments, the impedance matching circuit 115 isnot similar to the impedance matching circuit 114.

The impedance matching circuit 115 includes electrical components, e.g.,inductors, capacitors, etc., to match an impedance of a power sourcecoupled to the impedance matching circuit 115 with an impedance of aload coupled to the circuit 115. For example, the impedance matchingcircuit 114 matches an impedance of a source coupled to the impedancematching circuit 114, e.g., a combination of the m MHz RF generator, then MHz RF generator, and cables coupling the m and n MHz RF generators tothe impedance matching circuit 114, etc., with an impedance of a load,e.g., a combination of the plasma chamber 134 and the RF transmissionline 287, etc.

It should be noted that in an embodiment, the RF transmission line 287is similar to the RF transmission line 113 (FIG. 1). For example, animpedance of the RF transmission line 287 is the same as an impedance ofthe RF transmission line 113. As another example, an impedance of the RFtransmission line 287 is within a window, e.g., within 10-20%, of theimpedance of the RF transmission line 113. In various embodiments, theRF transmission line 287 is not similar to the RF transmission line 113.

The plasma chamber 134 includes an ESC 192, an upper electrode 264, andother parts (not shown), e.g., an upper dielectric ring surrounding theupper electrode 264, an upper electrode extension surrounding the upperdielectric ring, a lower dielectric ring surrounding a lower electrodeof the ESC 192, a lower electrode extension surrounding the lowerdielectric ring, an upper plasma exclusion zone (PEZ) ring, a lower PEZring, etc. The upper electrode 264 is located opposite to and facing theESC 192. A work piece 262, e.g., a semiconductor wafer, etc., issupported on an upper surface 263 of the ESC 192. Each of the upperelectrode 264 and the lower electrode of the ESC 192 is made of a metal,e.g., aluminum, alloy of aluminum, copper, etc.

In one embodiment, the upper electrode 264 includes a hole that iscoupled to a central gas feed (not shown). The central gas feed receivesone or more process gases from a gas supply (not shown). The upperelectrode 264 is grounded. The ESC 192 is coupled to the m MHz RFgenerator and the n MHz RF generator via the impedance matching circuit115.

When the process gas is supplied between the upper electrode 264 and theESC 192 and when the m MHz RF generator and/or the n MHz RF generatorsupplies power via the impedance matching circuit 115 to the ESC 192,the process gas is ignited to generate plasma within the plasma chamber134.

It should be noted that the system 128 lacks a probe, e.g., a metrologytool, a voltage and current probe, a voltage probe, etc., to measure thevariable at an output 283 of the impedance matching circuit 115, at apoint on the RF transmission line 287, or at the ESC 192. The values ofthe variable at the model nodes N1 m, N2 m, N4 m, and N6 m are used todetermine whether the system 128 is functioning as desired.

In various embodiments, the system 128 lacks a wafer bias sensor, e.g.,an in-situ direct current (DC) probe pick-up pin, and related hardwarethat is used to measure wafer bias at the ESC 192. The nonuse of thewafer bias sensor and the related hardware saves cost.

It should also be noted that in an embodiment, the system 128 includesany number of RF generators coupled to an impedance matching circuit.

FIGS. 12A, 12B, and 12C are diagrams of embodiments of graphs 268, 272,and 275 that illustrate a correlation between voltage, e.g., root meansquare (RMS) voltage, peak voltage, etc., that is measured at theoutput, e.g., the node N4, of the impedance matching circuit 114(FIG. 1) within the system 126 (FIG. 1) by using a voltage probe and avoltage, e.g., peak voltage, etc., at a corresponding model node output,e.g., the node N4 m, determined using the method 102 (FIG. 2). Moreover,FIGS. 12A, 12C, and 12E are diagrams of embodiments of graphs 270, 274,and 277 that illustrate a correlation between current, e.g., root meansquare (RMS) current, etc., that is measured the output, e.g., the nodeN4, of the system 126 (FIG. 1) by using a current probe and a current,e.g., RMS current, etc., at a corresponding output, e.g., the node N4 m,determined using the method 102 (FIG. 2).

The voltage determined using the method 102 is plotted on an x-axis ineach graph 268, 272, and 275 and the voltage measured with the voltageprobe is plotted on a y-axis in each graph 268, 272, and 275. Similarly,the current determined using the method 102 is plotted on an x-axis ineach graph 270, 227, and 277 and the current measured with the currentprobe is plotted on a y-axis in each graph 270, 274, and 277.

The voltages are plotted in the graph 268 when the x MHz RF generator ison and the y MHz RF generator and a z MHz RF generator, e.g., 60 MHz RFgenerator, are off. Moreover, the voltages are plotted in the graph 272when the y MHz RF generator is on and the x and z MHz RF generators areoff. Also, the voltages are plotted in the graph 275 when the z MHz RFgenerator is on and the x and y MHz RF generators are off.

Similarly, currents are plotted in the graph 270 when the x MHz RFgenerator is on and the y MHz RF generator and a z MHz RF generator areoff. Also, the currents are plotted in the graph 274 when the y MHz RFgenerator is on and the x and z MHz RF generators are off. Also, thecurrents are plotted in the graph 277 when the z MHz RF generator is onand the x and y MHz RF generators are off.

It can be seen in each graph 268, 272, and 275 that an approximatelylinear correlation exists between the voltage plotted on the y-axis ofthe graph and the voltage plotted on the x-axis of the graph. Similarly,it can be seen in each graph 270, 274, and 277 that an approximatelylinear correlation exists between the current plotted on the y-axis andthe current plotted on the x-axis.

FIG. 13 is a flowchart of an embodiment of the method 340 fordetermining wafer bias at a model node, e.g., the model node N4 m, themodel node N1 m, the model node N2 m, the model node N6 m, etc., of theplasma system 126 (FIG. 1). It should be noted that in some embodiments,wafer bias is a direct current (DC) voltage that is created by plasmagenerated within the plasma chamber 175 (FIG. 1). In these embodiments,the wafer bias is present on a surface, e.g., the upper surface 183, ofthe ESC 177 (FIG. 1) and/or on a surface, e.g., an upper surface, of thework piece 131 (FIG. 1).

It should further be noted that the model nodes N1 m and N2 m are on theRF transmission model 161 (FIG. 1) and the model node N6 m is on the ESCmodel 125 (FIG. 1). The method 340 is executed by the processor of thehost system 130 (FIG. 1). In the method 340, the operation 106 isperformed.

Moreover, in an operation 341, one or more models, e.g. the impedancematching model 104, the RF transmission model 161, the ESC model 125(FIG. 1), a combination thereof, etc., of corresponding one or moredevices, e.g., the impedance matching circuit 114, the RF transmissionline 113, the ESC 177, a combination thereof, etc., are generated. Forexample, the ESC model 125 is generated with similar characteristics tothat of the ESC 177 (FIG. 1).

In an operation 343, the complex voltage and current identified in theoperation 106 is propagated through one or more elements of the one ormore models to determine a complex voltage and current at an output ofthe one or more models. For example, the second complex voltage andcurrent is determined from the first complex voltage and current. Asanother example, the second complex voltage and current is determinedfrom the first complex voltage and current and the third complex voltageand current is determined from the second complex voltage and current.As yet another example, the second complex voltage and current isdetermined from the first complex voltage and current, the third complexvoltage and current is determined from the second complex voltage andcurrent, and the third complex voltage and current is propagated throughthe portion 197 of the RF transmission model 161 (FIG. 1) to determine afourth complex voltage and current at the model node N2 m. In thisexample, the fourth complex voltage and current is determined bypropagating the third complex voltage and current through impedances ofelements of the portion 197. As yet another example, the RF transmissionmodel 161 provides an algebraic transfer function that is executed bythe processor of the host system 130 to translate the complex voltageand current measured at one or more outputs of one or more RF generatorsto an electrical node, e.g., the model node N1 m, the model node N2 m,etc., along the RF transmission model 161.

As another example of the operation 343, the second complex voltage andcurrent is determined from the first complex voltage and current, thethird complex voltage and current is determined from the second complexvoltage and current, the fourth complex voltage and current isdetermined from the third complex voltage and current, and the fourthcomplex voltage and current is propagated through the ESC model 125 todetermine a fifth complex voltage and current at the model node N6 m. Inthis example, the fifth complex voltage and current is determined bypropagating the fourth complex voltage and current through impedances ofelements, e.g., capacitors, inductors, etc., of the ESC model 125.

In an operation 342, a wafer bias is determined at the output of the oneor more models based on a voltage magnitude of the complex voltage andcurrent at the output, a current magnitude of the complex voltage andcurrent at the output, and a power magnitude of the complex voltage andcurrent at the output. For example, wafer bias is determined based on avoltage magnitude of the second complex voltage and current, a currentmagnitude of the second complex voltage and current, and a powermagnitude of the second complex voltage and current. To furtherillustrate, when the x MHz RF generator is on and the y MHz and z MHz RFgenerators are off, the processor of the host system 130 (FIG. 1)determines wafer bias at the model node N4 m (FIG. 1) as a sum of afirst product, a second product, a third product, and a constant. Inthis illustration, the first product is a product of a first coefficientand the voltage magnitude of the second complex voltage and current, thesecond product is a product of a second coefficient and the currentmagnitude of the second complex voltage and current, and the thirdproduct is a product of a square root of a third coefficient and asquare root of a power magnitude of the second complex voltage andcurrent.

As an example, a power magnitude is a power magnitude of deliveredpower, which is determined by the processor of the host system 130 as adifference between forward power and reflected power. Forward power ispower supplied by one or more RF generators of the system 126 (FIG. 1)to the plasma chamber 175 (FIG. 1). Reflected power is power reflectedback from the plasma chamber 175 towards one or more RF generators ofthe system 126 (FIG. 1). As an example, a power magnitude of a complexvoltage and current is a determined by the processor of the host system130 as a product of a current magnitude of the complex voltage andcurrent and a voltage magnitude of the complex voltage and current.Moreover, each of a coefficient and a constant used to determine a waferbias is a positive or a negative number. As another example ofdetermination of the wafer bias, when the x MHz RF generator is on andthe y and z MHz RF generators are off, the wafer bias at a model node isrepresented as ax*Vx+bx*Ix+cx*sqrt (Px)+dx, where “ax” is the firstcoefficient, “bx” is the second coefficient, “dx” is the constant, “Vx”is a voltage magnitude of a complex voltage and current at the modelnode “Ix” is a current magnitude of the complex voltage and current atthe model node, and “Px” is a power magnitude of the complex voltage andcurrent at the model node. It should be noted that “sqrt” is a squareroot operation, which is performed by the processor of the host system130. In some embodiments, the power magnitude Px is a product of thecurrent magnitude Ix and the voltage magnitude Vx.

In various embodiments, a coefficient used to determine a wafer bias isdetermined by the processor of the host system 130 (FIG. 1) based on aprojection method. In the projection method, a wafer bias sensor, e.g.,a wafer bias pin, etc., measures wafer bias on a surface, e.g., theupper surface 183 (FIG. 1), etc., of the ESC 177 for a first time.Moreover, in the projection method, a voltage magnitude, a currentmagnitude, and a power magnitude are determined at a model node withinthe plasma system 126 based on complex voltage and current measured atan output of an RF generator. For example, the complex voltage andcurrent measured at the node N3 (FIG. 1) for the first time ispropagated by the processor of the host system 130 to a model node,e.g., the model node N4 m, the model node N1 m, the model node N2 m, orthe model node N6 m (FIG. 1), etc., to determine complex voltage andcurrent at the model node for the first time. Voltage magnitude andcurrent magnitude are extracted by the processor of the host system 130from the complex voltage and current at the model node for the firsttime. Also, power magnitude is calculated by the processor of the hostsystem 130 as a product of the current magnitude and the voltagemagnitude for the first time.

Similarly, in the example, complex voltage and current is measured atthe node N3 for one or more additional times and the measured complexvoltage and current is propagated to determine complex voltage andcurrent at the model node, e.g., the model node N4 m, the model node N1m, the model node N2 m, the model node N6 m, etc., for the one or moreadditional times. Also, for the one or more additional times, voltagemagnitude, current magnitude, and power magnitude are extracted from thecomplex voltage and current determined for the one or more additionaltimes. A mathematical function, e.g., partial least squares, linearregression, etc., is applied by the processor of the host system 130 tothe voltage magnitude, the current magnitude, the power magnitude, andthe measured wafer bias obtained for the first time and for the one ormore additional times to determine the coefficients ax, bx, cx and theconstant dx.

As another example of the operation 342, when the y MHz RF generator ison and the x and z MHz RF generators are off, a wafer bias is determinedas ay*Vy+by*Iy+cy*sqrt (Py)+dy, where “ay” is a coefficient, “by” is acoefficient, “dy” is a constant, “Vy” is a voltage magnitude of thesecond complex voltage and current, “Iy” is a current magnitude of thesecond complex voltage and current, and “Px” is a power magnitude of thesecond complex voltage and current. The power magnitude Py is a productof the current magnitude Iy and the voltage magnitude Vy. As yet anotherexample of the operation 342, when the z MHz RF generator is on and thex and y MHz RF generators are off, a wafer bias is determined asaz*Vz+bz*Iz+cz*sqrt (Pz)+dz, where “az” is a coefficient, “bz” is acoefficient, “dz” is a constant, “Vz” is a voltage magnitude of thesecond complex voltage and current, “Iz” is a current magnitude of thesecond complex voltage and current, and “Pz” is a power magnitude of thesecond complex voltage and current. The power magnitude Pz is a productof the current magnitude Iz and the voltage magnitude Vz.

As another example of the operation 342, when the x and y MHz RFgenerators are on and the z MHz RF generator is off, the wafer bias isdetermined as a sum of a first product, a second product, a thirdproduct, a fourth product, a fifth product, a sixth product, and aconstant. The first product is a product of a first coefficient and thevoltage magnitude Vx, the second product is a product of a secondcoefficient and the current magnitude Ix, the third product is a productof a third coefficient and a square root of the power magnitude Px, thefourth product is a product of a fourth coefficient and the voltagemagnitude Vy, the fifth product is a product of a fifth coefficient andthe current magnitude Iy, and the sixth product is a product of a sixthcoefficient and a square root of the power magnitude Py. When the x andy MHz RF generators are on and the z MHz RF generator is off, the waferbias is represented as axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, where “axy”, “bxy”, “cxy”, “dxy”,“exy”, “fxy”, “dxy”, “exy”, and “fxy” are coefficients, and “gxy” is aconstant.

As another example of the operation 342, when the y and z MHz RFgenerators are on and the x MHz RF generator is off, a wafer bias isdetermined as ayz*Vy+byz*Iy+cyz*sqrt (Py)+dyz*Vz+eyz*Iz+fyz*sqrt(Pz)+gyz, where “ayz”, “byz”, “cyz”, “dyz”, “eyz”, and “fyz” arecoefficients, and “gyz” is a constant. As yet another example of theoperation 342, when the x and z MHz RF generators are on and the y MHzRF generator is off, a wafer bias is determined asaxz*Vx+bxz*Ix+cxz*sqrt (Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, where“axz”, “bxz”, “cxz”, “dxz”, “exz”, and “fxz” are coefficients, and gxzis a constant.

As another example of the operation 342, when the x, y, and z MHz RFgenerators are on, the wafer bias is determined as a sum of a firstproduct, a second product, a third product, a fourth product, a fifthproduct, a sixth product, a seventh product, an eighth product, a ninthproduct, and a constant. The first product is a product of a firstcoefficient and the voltage magnitude Vx, the second product is aproduct of a second coefficient and the current magnitude Ix, the thirdproduct is a product of a third coefficient and a square root of thepower magnitude Px, the fourth product is a product of a fourthcoefficient and the voltage magnitude Vy, the fifth product is a productof a fifth coefficient and the current magnitude Iy, the sixth productis a product of a sixth coefficient and a square root of the powermagnitude Py, the seventh product is a product of a seventh coefficientand the voltage magnitude Vz, the eighth product is a product of aneighth coefficient and the current magnitude Iz, and the ninth productis a product of a ninth coefficient and a square root of a powermagnitude Pz. When the x, y, and z MHz RF generators are on, the waferbias is represented as axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyz,where “axyz”, “bxyz”, “cxyz”, “dxyz”, “exyz”, “fxyz”, “gxyz”, “hxyz”,and “ixyz” are coefficients, and “jxyz” is a constant.

As another example of determination of wafer bias at the output of theone or more models, wafer bias at the model node N1 m is determined bythe processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N1 m. To further illustrate, thesecond complex voltage and current is propagated along the portion 173(FIG. 1) to determine complex voltage and current at the model node N1m. The complex voltage and current is determined at the model node N1 mfrom the second complex voltage and current in a manner similar to thatof determining the second complex voltage and current from the firstcomplex voltage and current. For example, the second complex voltage andcurrent is propagated along the portion 173 based on characteristics ofelements of the portion 173 to determine a complex voltage and currentat the model node N1 m.

Based on the complex voltage and current determined at the model node N1m, wafer bias is determined at the model node N1 m by the processor ofthe host system 130. For example, wafer bias is determined at the modelnode N1 m from the complex voltage and current at the model node N1 m ina manner similar to that of determining the wafer bias at the model nodeN4 m from the second complex voltage and current. To illustrate, whenthe x MHz RF generator is on and the y MHz and z MHz RF generators areoff, the processor of the host system 130 (FIG. 1) determines wafer biasat the model node N1 m as a sum of a first product, a second product, athird product, and a constant. In this example, the first product is aproduct of a first coefficient and the voltage magnitude of the complexvoltage and current at the model node N1 m, the second product is aproduct of a second coefficient and the current magnitude of the complexvoltage and current at the model node N1 m, and the third product is aproduct of a square root of a third coefficient and a square root of apower magnitude of the complex voltage and current at the model node N1m. When the x MHz RF generator is on and the y and z MHz RF generatorsare off, the wafer bias at the model node N1 m is represented asax*Vx+bx*Ix+cx*sqrt (Px)+dx, where ax is the first coefficient, bx isthe second coefficient, cx is the third coefficient, dx is the constant,Vx is the voltage magnitude at the model node N1 m, Ix is the currentmagnitude at the model node N1 m, and Px is the power magnitude at themodel node N1 m.

Similarly, based on the complex voltage and current at the model node N1m and based on which of the x, y, and z MHz RF generators are on, thewafer bias ay*Vy+by*Iy+cy*sqrt (Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz,axy*Vx+bxy*Ix+cxy*sqrt (Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy,axz*Vx+bxz*Ix+cxz*sqrt (Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz,ayz*Vy+byz*Iy+cyz*sqrt (Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, andaxyz*Vx+bxyz*Ix+cxyz*sqrt (Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt(Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyz are determined.

As yet another example of determination of wafer bias at the output ofthe one or more models, wafer bias at the model node N2 m is determinedby the processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N2 m in a manner similar to thatof determining wafer bias at the model node N1 m based on voltage andcurrent magnitudes determined at the model node N1 m. To furtherillustrate, wafer bias ax*Vx+bx*Ix+cx*sqrt (Px)+dx, ay*Vy+by*Iy+cy*sqrt(Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz, axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, axz*Vx+bxz*Ix+cxz*sqrt(Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, ayz*Vy+byz*Iy+cyz*sqrt(Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, and axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyzare determined at the model node N2 m.

As another example of determination of wafer bias at the output of theone or more models, wafer bias at the model node N6 m is determined bythe processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N6 m in a manner similar to thatof determining wafer bias at the model node N2 m based on voltage andcurrent magnitudes determined at the model node N2 m. To furtherillustrate, wafer bias ax*Vx+bx*Ix+cx*sqrt (Px)+dx, ay*Vy+by*Iy+cy*sqrt(Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz, axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, axz*Vx+bxz*Ix+cxz*sqrt(Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, ayz*Vy+byz*Iy+cyz*sqrt(Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, and axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyzare determined at the model node N6 m.

It should be noted that in some embodiments, wafer bias is stored withinthe storage HU 162 (FIG. 1).

FIG. 14 is a state diagram illustrating an embodiment of a wafer biasgenerator 340, which is implemented within the host system 130 (FIG. 1).When all of the x, y, and z MHz RF generators are off, wafer bias iszero or minimal at a model node, e.g., the model node N4 m, N1 m, N2 m,N6 m (FIG. 1), etc. When the x, y, or z MHz RF generator is on and theremaining of the x, y, and z MHz RF generators are off, the wafer biasgenerator 340 determines a wafer bias at a model node, e.g., the modelnode N4 m, N1 m, N2 m, N6 m, etc., as a sum of a first product a*V, asecond product b*I, a third product c*sqrt(P), and a constant d, where Vis a voltage magnitude of a complex voltage and current at the modelnode, I is a current magnitude of the complex voltage and current, P isa power magnitude of the complex voltage and current, a is acoefficient, b is a coefficient, c is a coefficient, and d is aconstant. In various embodiments, a power magnitude at a model node is aproduct of a current magnitude at the model node and a voltage magnitudeat the model node. In some embodiments, the power magnitude is amagnitude of delivered power.

When two of the x, y, and z MHz RF generators are on and the remainingof the x, y, and z MHz RF generators are off, the wafer bias generator340 determines a wafer bias at a model node, e.g., the model node N4 m,N1 m, N2 m, N6 m, etc., as a sum of a first product a12*V1, a secondproduct b12*I1, a third product c12*sqrt(P1), a fourth product d12*V2, afifth product e12*I2, a sixth product f12*sqrt(P2), and a constant g12,where “V1” is a voltage magnitude of a complex voltage and current atthe model node determined by propagating a voltage measured at an outputof a first one of the RF generators that is on, “I1” is a currentmagnitude of the complex voltage and current determined by propagating acurrent measured at the output of the first RF generator that is on,“P1” is a power magnitude of the complex voltage and current determinedas a product of V1 and I1, “V2” is a voltage magnitude of the complexvoltage and current at the model node determined by propagating avoltage measured at an output of a second one of the RF generators thatis on, “I2” is a current magnitude of the complex voltage and currentdetermined by propagating the current measured at an output of thesecond RF generator that is on, “P2” is a power magnitude determined asa product of V2 and I2, each of “a12”, “b12”, “c12”, “d12”, “e12” and“f12” is a coefficient, and “g12” is a constant.

When all of the x, y, and z MHz RF generators are on, the wafer biasgenerator 340 determines a wafer bias at a model node, e.g., the modelnode N4 m, N1 m, N2 m, N6 m, etc., as a sum of a first product a123*V1,a second product b123*I1, a third product c123*sqrt(P1), a fourthproduct d123*V2, a fifth product e123*I2, a sixth product f123*sqrt(P2),a seventh product g123*V3, an eighth product h123*I3, a ninth producti123*sqrt(P3), and a constant j123, where “V1” is a voltage magnitude ofa complex voltage and current at the model node determined bypropagating a voltage measured at an output of a first one of the RFgenerators, “I1” is a current magnitude of the complex voltage andcurrent determined by propagating a current measured at the output ofthe first RF generator, “P1” is a power magnitude of the complex voltageand current determined as a product of V1 and I1, “V2” is a voltagemagnitude of the complex voltage and current at the model nodedetermined by propagating a voltage measured at an output of a secondone of the RF generators, “I2” is a current magnitude of the complexvoltage and current determined by propagating a current measured at theoutput of the second RF generator, “P2” is a power magnitude of thecomplex voltage and current determined as a product of V2 and I2, “V3”is a voltage magnitude of the complex voltage and current at the modelnode determined by propagating a voltage measured at an output of athird one of the RF generators, “I3” is a current magnitude of thecomplex voltage and current determined by propagating a current at theoutput of the third RF generator, “P3” is a power magnitude of thecomplex voltage and current determined as a product of V3 and I3, eachof “a123”, “b123”, “c123”, “d123”, “e123”, “f123”, “g123”, “h123”, and“i123” is a coefficient, and “j123” is a constant.

FIG. 15 is a flowchart of an embodiment of the method 351 fordetermining a wafer bias at a point along a path 353 (FIG. 16) betweenthe model node N4 m (FIG. 16) and the ESC model 125 (FIG. 16). FIG. 15is described with reference to FIG. 16, which is a block diagram of anembodiment of a system 355 for determining a wafer bias at an output ofa model.

In an operation 357, output of the x, y, or z MHz RF generator isdetected to identify a generator output complex voltage and current. Forexample, the voltage and current probe 110 (FIG. 1) measures complexvoltage and current at the node N3 (FIG. 1). In this example, thecomplex voltage and current is received from the voltage and currentprobe 110 via the communication device 185 (FIG. 1) by the host system130 (FIG. 1) for storage within the storage HU 162 (FIG. 1). Also, inthe example, the processor of the host system 130 identifies the complexvoltage and current from the storage HU 162.

In an operation 359, the processor of the host system 130 uses thegenerator output complex voltage and current to determine a projectedcomplex voltage and current at a point along the path 353 between themodel node N4 m and the model node N6 m. The path 161 extends from themodel node N4 m to the model node N6 m. For example, the fifth complexvoltage and current is determined from the complex voltage and currentmeasured at the output of the x MHz RF generator, the y MHz RFgenerator, or the z MHz RF generator. As another example, the complexvoltage and current measured at the node N3 or the node N5 is propagatedvia the impedance matching model 104 to determine a complex voltage andcurrent at the model node N4 m (FIG. 1). In the example, the complexvoltage and current at the model node N4 m is propagated via one or moreelements of the RF transmission model 161 (FIG. 16) and/or via one ormore elements of the ESC model 125 (FIG. 16) to determine complexvoltage and current at a point on the path 353.

In an operation 361, the processor of the host system 130 applies theprojected complex voltage and current determined at the point on thepath 353 as an input to a function to map the projected complex voltageand current to a wafer bias value at the node N6 m of the ESC model 125(FIG. 15). For example, when the x, y, or z MHz RF generator is on, awafer bias at the model node N6 m is determined as a sum of a firstproduct a*V, a second product b*I, a third product c*sqrt(P), and aconstant d, where, V is a voltage magnitude of the projected complexvoltage and current at the model node N6 m, I is a current magnitude ofthe projected complex voltage and current at the model node N6 m, P is apower magnitude of the projected complex voltage and current at themodel node N6 m, a, b, and c are coefficients, and d is a constant.

As another example, when two of the x, y, and z MHz RF generators are onand the remaining of the x, y, and z MHz RF generators are off, a waferbias at the model node N6 m is determined as a sum of a first producta12*V1, a second product b12*I1, a third product c12*sqrt(P1), a fourthproduct d12*V2, a fifth product e12*I2, a sixth product f12*sqrt(P2),and a constant g12, where V1 is a voltage magnitude at the model node N6m as a result of a first one of the two RF generators being on, I1 is acurrent magnitude at the model node N6 m as a result of the first RFgenerator being on, P1 is a power magnitude at the model node N6 m as aresult of the first RF generator being on, V2 is a voltage magnitude atthe model node N6 m as a result of a second one of the two RF generatorsbeing on, I2 is a current magnitude at the model node N6 m as a resultof the second RF generator being on, and P2 is a power magnitude at themodel node N6 m as a result of the second RF generator being on, a12,b12, c12, d12, e12, and f12 are coefficients, and g12 is a constant.

As yet another example, when all of the x, y, and z MHz RF generatorsare on, a wafer bias at the model node N6 m is determined as a sum of afirst product a123*V1, a second product b123*I1, a third productc123*sqrt(P1), a fourth product d123*V2, a fifth product e123*I2, asixth product f123*sqrt(P2), a seventh product g123*V3, an eighthproduct h123*I3, a ninth product i123*sqrt(P3), and a constant j123,where V1, I1, P1, V2, I2, and P2 are described above in the precedingexample, V3 is a voltage magnitude at the model node N6 m as a result ofa third one of the RF generators being on, I3 is a current magnitude atthe model node N6 m as a result of the third RF generator being on, andP3 is a power magnitude at the model node N6 m as a result of the thirdRF generator being on, a123, b123, c123, d123, e123, f123, g123, h123,and i123 are coefficients and j123 is a constant.

As another example, a function used to determine a wafer bias is a sumof characterized values and a constant. The characterized values includemagnitudes, e.g., the magnitudes V, I, P, V1, I1, P1, V2, I2, P2, V3,I3, P3, etc. The characterized values also include coefficients, e.g.,the coefficients, a, b, c, a12, b12, c12, d12, e12, f12, a123, b123,c123, d123, e123, f123, g123, h123, i123, etc. Examples of the constantinclude the constant d, the constant g12, the constant j123, etc.

It should be noted that the coefficients of the characterized values andthe constant of the characterized values incorporate empirical modelingdata. For example, wafer bias is measured for multiple times at the ESC177 (FIG. 1) using a wafer bias sensor. Moreover, in the example, forthe number of times the wafer bias is measured, complex voltages andcurrents at the point along the path 353 (FIG. 16) are determined bypropagating the complex voltage and current from one or more of thenodes, e.g., the nodes N3, N5, etc., of one or more of the RFgenerators, e.g., the x MHz RF generator, the y MHz RF generator, the zMHz RF generator, etc., via one or more of the models, e.g., theimpedance matching model 104, the model portion 173, the RF transmissionmodel 161, the ESC model 125 (FIG. 1), to reach to the point on the path353 (FIG. 16). Moreover, in this example, a statistical method, e.g.,partial least squares, regression, etc., is applied by the processor ofthe host system 130 to the measured wafer bias and to voltagemagnitudes, current magnitudes, and power magnitudes extracted from thecomplex voltages and currents at the point to determine the coefficientsof the characterized values and the constant of the characterizedvalues.

In various embodiments, a function used to determine a wafer bias ischaracterized by a summation of values that represent physicalattributes of the path 353. The physical attributes of the path 353 arederived values from test data, e.g., empirical modeling data, etc.Examples of physical attributes of the path 353 include capacitances,inductances, a combination thereof, etc., of elements on the path 353.As described above, the capacitances and/or inductances of elementsalong the path 353 affect voltages and currents empirically determinedusing the projection method at the point on the path 353 and in turn,affect the coefficients of the characterized values and the constant ofthe characterized values.

In some embodiments, a function used to determine a wafer bias is apolynomial.

FIG. 17 is a flowchart of an embodiment of the method 363 fordetermining a wafer bias at a model node of the system 126 (FIG. 1).FIG. 17 is described with reference to FIGS. 1 and 16. The method 363 isexecuted by the processor of the host system 130 (FIG. 1). In anoperation 365, one or more complex voltages and currents are received bythe host system 130 from one or more communication devices of agenerator system, which includes one or more of the x MHz RF generator,the y MHz RF generator, and the z MHz RF generator. For example, complexvoltage and current measured at the node N3 is received from thecommunication device 185 (FIG. 1). As another example, complex voltageand current measured at the node N5 is received from the communicationdevice 189 (FIG. 1). As yet another example, complex voltage and currentmeasured at the node N3 and complex voltage and current measured at thenode N5 are received. It should be noted that an output of the generatorsystem includes one or more of the nodes N3, N5, and an output node ofthe z MHz RF generator.

In an operation 367, based on the one or more complex voltages andcurrents at the output of the generator system, a projected complexvoltage and current is determined at a point along, e.g., on, etc., thepath 353 (FIG. 16) between the impedance matching model 104 and the ESCmodel 125 (FIG. 16). For example, the complex voltage and current at theoutput of the generator system is projected via the impedance matchingmodel 104 (FIG. 16) to determine a complex voltage and current at themodel node N4 m. As another example, the complex voltage and current atthe output of the generator system is projected via the impedancematching model 104 and the portion 173 (FIG. 1) of the RF transmissionmodel 161 to determine a complex voltage and current at the model nodeN1 m (FIG. 1). As yet another example, the complex voltage and currentat the output of the generator system is projected via the impedancematching model 104 and the RF transmission model 161 to determine acomplex voltage and current at the model node N2 m (FIG. 1). As anotherexample, the complex voltage and current at the output of the generatorsystem is projected via the impedance matching model 104, the RFtransmission model 161, and the ESC model 125 to determine a complexvoltage and current at the model node N6 m (FIG. 1).

In an operation 369, a wafer bias is calculated at the point along thepath 353 by using the projected complex V&I as an input to a function.For example, when the x, y, or z MHz RF generator is on and theremaining of the x, y, and z MHz RF generators are off, a wafer bias atthe point is determined from a function, which is as a sum of a firstproduct a*V, a second product b*I, a third product c*sqrt(P), and aconstant d, where, V is a voltage magnitude of the projected complexvoltage and current at the point, I is a current magnitude of theprojected complex voltage and current at the point, P is a powermagnitude of the projected complex voltage and current at the point, a,b, and c are coefficients, and d is a constant.

As another example, when two of the x, y, and z MHz RF generators are onand the remaining of the x, y, and z MHz RF generators are off, a waferbias at the point is determined as a sum of a first product a12*V1, asecond product b12*I1, a third product c12*sqrt(P1), a fourth productd12*V2, a fifth product e12*I2, a sixth product f12*sqrt(P2), and aconstant g12, where V1 is a voltage magnitude at the point as a resultof a first one of the two RF generators being on, I1 is a currentmagnitude at the point as a result of the first RF generator being on,P1 is a power magnitude at the point as a result of the first RFgenerator being on, V2 is a voltage magnitude at the point as a resultof a second one of the two RF generators being on, I2 is a currentmagnitude at the point as a result of the second RF generator being on,and P2 is a power magnitude at the point as a result of the second RFgenerator being on, a12, b12, c12, d12, e12, and f12 are coefficients,and g12 is a constant.

As yet another example, when all of the x, y, and z MHz RF generatorsare on, a wafer bias at the point is determined as a sum of a firstproduct a123*V1, a second product b123*I1, a third productc123*sqrt(P1), a fourth product d123*V2, a fifth product e123*I2, asixth product f123*sqrt(P2), a seventh product g123*V3, an eighthproduct h123*I3, a ninth product i123*sqrt(P3), and a constant j123,where V1, I1, P1, V2, I2, and P2 are described above in the precedingexample, V3 is a voltage magnitude at the point as a result of a thirdone of the RF generators being on, I3 is a current magnitude at thepoint as a result of the third RF generator being on, and P3 is a powermagnitude at the point as a result of the third RF generator being on,a123, b123, c123, d123, e123, f123, g123, h123, and i123 arecoefficients, and j123 is a constant.

FIG. 18 is a block diagram of an embodiment of a system 330 that is usedto illustrate advantages of determining wafer bias by using the method340 (FIG. 13), the method 351 (FIG. 15), or the method 363 (FIG. 17)instead of by using a voltage probe 332, e.g., a voltage sensor, etc.

The voltage probe 332 is coupled to the node N1 to determine a voltageat the node N1. In some embodiments, the voltage probe 332 is coupled toanother node, e.g., node N2, N4, etc., to determine voltage at the othernode. The voltage probe 332 includes multiple circuits, e.g., an RFsplitter circuit, a filter circuit 1, a filter circuit 2, a filtercircuit 3, etc.

Also, the x and y MHz RF generators are coupled to a host system 334that includes a noise or signal determination module 336. It should benoted that a module may be a processor, an ASIC, a PLD, a softwareexecuted by a processor, or a combination thereof.

The voltage probe 332 measures a voltage magnitude, which is used by thehost system 334 to determine a wafer bias. The module 336 determineswhether the voltage magnitude measured by the voltage probe 332 is asignal or noise. Upon determining that the voltage magnitude measured bythe voltage probe 332 is a signal, the host system 334 determines waferbias.

The system 126 (FIG. 1) is cost effective compared to the system 330 andsaves time and effort compared to the system 330. The system 330includes the voltage probe 332, which does not need to be included inthe system 126. There is no need to couple a voltage probe at the nodeN4, N1, or N2 of the system 126 to determine wafer bias. In the system126, wafer bias is determined based on the impedance matching model 104,RF transmission model 161, and/or the ESC model 125 (FIG. 1). Moreover,the system 330 includes the module 336, which also does not need to beincluded in the system 126. There is no need to spend time and effort todetermine whether a complex voltage and current is a signal or noise. Nosuch determination needs to be made by the host system 130 (FIG. 1).

FIGS. 19A, 19B, and 19C show embodiments of graphs 328, 332, and 336 toillustrate a correlation, e.g., a linear correlation, etc., betweenvoltage, e.g., peak voltage, etc., that is measured at the output, e.g.,the node N1, of the portion 195 (FIG. 1) by using a voltage probe and avoltage, e.g., peak voltage, etc., at a corresponding model node output,e.g., the node N1 m, determined using the method 102 (FIG. 2). In eachgraph 328, 332, and 336, the measured voltage is plotted on a y-axis andthe voltage determined using the method 102 is plotted on an x-axis.

Moreover, FIGS. 19A, 19B, and 19C show embodiments of graphs 330, 334,and 338 to illustrate a correlation, e.g., a linear correlation, etc.,between wafer bias that is measured at the output N6 (FIG. 1) by using awafer bias probe and wafer bias at a corresponding model node output,e.g., the node N6 m, determined using the method 340 (FIG. 13), themethod 351 (FIG. 15), or the method 363 (FIG. 17). In each graph 330,334, and 338, the wafer bias determined using the wafer bias probe isplotted on a y-axis and the wafer bias determined using the method 340,the method 351, or the method 363 is plotted on an x-axis.

The voltages and wafer bias are plotted in the graphs 328 and 330 whenthe y MHz and z MHz RF generators are on and the x MHz RF generator isoff. Moreover, the voltages and wafer bias are plotted in the graphs 332and 334 when the x MHz and z MHz RF generators are on and the y MHz RFgenerator is off. Also, the voltages are plotted in the graphs 336 and338 when the x MHz and y MHz RF generators are on and the z MHz RFgenerator is off.

FIG. 20A is a diagram of an embodiment of graphs 276 and 278 toillustrate that there is a correlation between a wired wafer biasmeasured using a sensor tool, e.g., a metrology tool, a probe, a sensor,a wafer bias probe, etc., a model wafer bias that is determined usingthe method 340 (FIG. 13), the method 351 (FIG. 15), or the method 363(FIG. 17), and an error in the model bias. The wired wafer bias that isplotted in the graph 276 is measured at a point, e.g., a node on the RFtransmission line 113, a node on the upper surface 183 (FIG. 1) of theESC 177, etc. and the model bias that is plotted in the graph 276 isdetermined at the corresponding model point, e.g., the model node N4 m,the model node N1 m, the model node N2 m, the model node N6 m, etc.(FIG. 1), on the path 353 (FIG. 16). The wired wafer bias is plottedalong a y-axis in the graph 276 and the model bias is plotted along anx-axis in the graph 276.

The wired wafer bias and the model bias are plotted in the graph 276when the x MHz RF generator is on, and the y and z MHz RF generators areoff. Moreover, the model bias of graph 276 is determined using anequation a2*V2+b2*I2+c2*sqrt (P2)+d2, where “*” representsmultiplication, “sqrt” represents a square root, “V2” represents voltageat the point along the path 353 (FIG. 16), I2 represents current at thepoint, P2 represents power at the point, “a2” is a coefficient, “b2” isa coefficient, “c2” is a coefficient, and “d2” is a constant value.

The graph 278 plots an error, which is an error in the model bias at thepoint, on a y-axis and plots the model bias at the point on an x-axis.The model error is an error, e.g., a variance, a standard deviation,etc., in the model bias. The model error and the model bias are plottedin the graph 278 when the x MHz RF generator is on and the y and z MHzRF generators are off.

FIG. 20B is a diagram of an embodiment of graphs 280 and 282 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 280 and 282 are plotted in a manner similar to the graphs 276and 278 (FIG. 17A) except that the graphs 280 and 282 are plotted whenthe y MHz RF generator is on and the x and z MHz RF generators are off.Moreover, the model bias of the graphs 280 and 282 is determined usingan equation a27*V27+b27*I27+c27*sqrt (P27)+d27, where “V27” represents avoltage magnitude at the point along the path 353 (FIG. 16), “I27”represents a current magnitude at the point, “P27” represents a powermagnitude at the point, “a27” is a coefficient, “b27” is a coefficient,“c27” is a coefficient, and “d27” is a constant value.

FIG. 20C is a diagram of an embodiment of graphs 284 and 286 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 284 and 286 are plotted in a manner similar to the graphs 276and 278 (FIG. 17A) except that the graphs 284 and 286 are plotted whenthe z MHz RF generator is on and the x and y MHz RF generators are off.Moreover, the model bias of the graphs 284 and 286 is determined usingan equation a60*V60+b60*I60+c60*sqrt (P60)+d60, where “V60” represents avoltage magnitude at the point along the path 353 (FIG. 16), “I60”represents a current magnitude at the point, “P60” represents a powermagnitude at the point, “a60” is a coefficient, “b60” is a coefficient,“c60” is a coefficient, and “d60” is a constant value.

FIG. 20D is a diagram of an embodiment of graphs 288 and 290 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 288 and 290 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 288 and 290 are plotted whenthe x and y MHz RF generators are on, and the z MHz RF generator is off.Moreover, the model bias of the graphs 288 and 290 is determined usingan equation a227*V2+b227*I2+c227*sqrt (P2)+d227*V27+e227*I27+f227*sqrt(P27)+g227, where “a227”, “b227” and “c227”, “d227”, “e227” and “f227”are coefficients, and “g227” is a constant value.

FIG. 20E is a diagram of an embodiment of graphs 292 and 294 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 292 and 294 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 292 and 294 are plotted whenthe x and z MHz RF generators are on, and the y MHz RF generator is off.Moreover, the model bias of the graphs 292 and 294 is determined usingan equation a260*V2+b260*I2+c260*sqrt (P2)+d20*V60+e260*I60+f260*sqrt(P60)+g260, where “a260”, “b260” “c260”, “d260”, “e260” and “f260” arecoefficients, and “g260” is a constant value.

FIG. 20F is a diagram of an embodiment of graphs 296 and 298 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 296 and 298 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 296 and 298 are plotted whenthe y and z MHz RF generators are on, and the x MHz RF generator is off.Moreover, the model bias of the graphs 296 and 298 is determined usingan equation a2760*V27+b2760*I27+c2760*sqrt(P27)+d2760*V60+e2760*I60+f2760*sqrt (P60)+g2760, where “a2760”, “b2760”“c2760”, “d2760”, “e2760” and “f2760” are coefficients, and “g2760” is aconstant value.

FIG. 20G is a diagram of an embodiment of graphs 302 and 304 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 302 and 304 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 302 and 304 are plotted whenthe x, y and z MHz RF generators are on. Moreover, the model bias of thegraphs 302 and 304 is determined using an equationa22760*V2+b22760*I2+c22760*sqrt (P2)+d22760*V60+e22760*I60+f22760*sqrt(P60)+g22760*V27+h22760*I27+i22760*sqrt (P27)+j22760, where “a22760”,“b22760”, “c22760”, “d22760”, “e22760”, “f22760” “g22760”, “h22760”, and“i22760” are coefficients and “j22760” is a constant value.

FIG. 21 is a block diagram of an embodiment of the host system 130. Thehost system 130 includes a processor 168, the storage HU 162, an inputHU 380, an output HU 382, an input/output (I/O) interface 384, an I/Ointerface 386, a network interface controller (NIC) 388, and a bus 390.The processor 168, the storage HU 162, the input HU 380, the output HU382, the I/O interface 384, the I/O interface 386, and the NIC 388 arecoupled with each other via a bus 392. Examples of the input HU 380include a mouse, a keyboard, a stylus, etc. Examples of the output HU382 include a display, a speaker, or a combination thereof. The displaymay be a liquid crystal display, a light emitting diode display, acathode ray tube, a plasma display, etc. Examples of the NIC 388 includea network interface card, a network adapter, etc.

Examples of an I/O interface include an interface that providescompatibility between pieces of hardware coupled to the interface. Forexample, the I/O interface 384 converts a signal received from the inputHU 380 into a form, amplitude, and/or speed compatible with the bus 392.As another example, the I/O interface 386 converts a signal receivedfrom the bus 392 into a form, amplitude, and/or speed compatible withthe output HU 382.

Improving Accuracy of RF Transmission System Models for SelectedPortions of an RF Transmission Line

FIG. 22 is a block diagram of the RF transmission system 2200, inaccordance with an embodiment described in the present disclosure. TheRF transmission system 2200 includes one or more RF generators 2202. Therespective outputs of the RF generators 2202 are coupled to inputs ofcorresponding match circuits 2206 by an RF feed part one 2204 (e.g. RFtunnel). The outputs of the match circuit(s) 2206 are coupled by RF feedpart two 2208 (e.g., an RF strap) and part three 2210 (e.g., acylindrical RF feed) to an electrostatic chuck 2220 disposed in theprocess chamber 2218. A test probe 2240 such as an RF probe or othervoltage or current probe is often connected to the output of the RFgenerator 2202 to monitor the output of the RF generator. Avoltage/current (V/I) probe 2221 is included in the electrostatic chuckto measure the RF induced DC bias voltage that is induced on a wafer2223 by the RF in the plasma 2218A.

A controller 2222 includes recipe logic 2224 including the calibrationrecipe 2224A and other recipes 2224B for processing wafers in theprocess chamber 2218. The controller 2222 is coupled to the one or moreRF generators 2202 and provides the respective one or more RF controlsignals to each RF generator. The controller 2222 can also includeadditional operational recipes and logic 2226.

The RF transmission system 2200 is shown divided into eight stages2230-2239. At least a portion of the eight stages 2230-2239 cancorrespond to test points in the RF transmission system or separatecomponents (e.g., RF generator, 2202, match circuit 2206, RF feed,electrostatic chuck 2220). Alternatively, the stages 2230-2239 cancorrespond to selected electrical components within one or more of theRF generator, 2202, match circuit 2206, RF feed, electrostatic chuck2220.

As shown, a first stage includes the RF generator 2202. A second stagebegins at first test point at the output 2230 of the RF generator 2202and includes RF feed part one 2204 (e.g., an RF tunnel) extending to asecond test point at the input 2231 of the match circuit 2206. The matchcircuit 2206 including the internal resistive, capacitive and inductivecomponents forms the third stage with a third test point at the outputof the match circuit.

A fourth stage of the RF transmission system 2200 begins at the output2232 of the match circuit 2206 and includes the RF feed part two 2208(e.g., an RF strap) and continues to a fourth test point 2233 at anintermediate point in the RF feed such as where the RF strap connects tothe RF feed part three 2210 (e.g., circumferential RF feed). A fifthstage begins at the fourth test point 2233 and includes the RF feed partthree 2210 (e.g., circumferential RF feed) and continues to a fifth testpoint at an input 2234 of the electrostatic chuck 2220.

The electrostatic chuck 2220 and the components and structures thereinform the sixth stage which ends at the top surface 2236 of theelectrostatic chuck. The seventh stage 2238 includes the plasma 2218Aand the eighth stage 2239 includes the RF return path 2290. As discussedelsewhere herein, the dynamic nature of the plasma 2218A is verydifficult to accurately model.

Various instruments can detect dynamic parameters of the process such asRF induced DC bias voltage that is induced into the wafer 2223 by the RFpresent in the plasma 2218A. It should be understood that dividing theRF transmission system 2200 into eight stages is merely an example fordescription purposes and the RF transmission system can be divided intomore or fewer than eight stages. By way of example, the probe 2240 canmonitor the output of the RF generator 2202 and couple the measuredoutput to the controller 2222 or other sources such as an externalmonitoring system (not shown).

Each of stages 2230-2235 of the RF transmission system 2200 can besimplified to an equivalent series RLC and shut RLC circuits. Thecorresponding RLC values can then be used to create a mathematical modelof the selected portion of the RF transmission system, as described inmore detail above. The RF transmission line model can then be used topredict an output corresponding to a given input of each stage of the RFtransmission system 2200. The equivalent series RLC and shunt RLCcircuits of each of the stages 2230-2235 can be verified empirically bymeasuring actual inputs and outputs of each stage and comparing themeasured values to the values predicted by the equivalent series RLC andshut RLC circuits. By way of example, the probe 2240 and the V/I probe2221 and other instruments can monitor the actual RF transmission system2200 during operational processes such as during a configuration recipeto confirm the accuracy of the various values at the corresponding testpoints that were predicted RF transmission line model. As the accuracyof each modeled stage 2230-2235 of the RF transmission system isconfirmed, the overall accuracy of the RF transmission line model of theRF transmission system, from the RF generator 2202 through to the inputof the electrostatic chuck 2210 is increased.

FIG. 23 is a simplified diagram of the electrostatic chuck 2220, inaccordance with an embodiment described in the present disclosure.Unfortunately, the electrostatic chuck 2220 is very complex structureincluding multiple non-conductive layers 2220A, multiple conductivestructures 2220B, lift pins 2220D, conductive cables 2220E, instrumentssuch as V/I probe 2221 and in some instances active components such asheaters 2220C and cooling channels and chambers. As a result of thephysical complexity of the electrostatic chuck 2220, it is verydifficult to simplify the electrostatic chuck to an equivalent seriesRLC and shunt RLC circuits. It is theoretically possible to calculate anRF model of the electrostatic chuck 2220 based on the equivalent seriesRLC and shunt RLC circuit, however, the computational requirements arevery large and time consuming and thus are relatively impractical foruse for monitoring and controlling plasma 2218A within the plasmaprocess chamber 2218.

Further, the plasma processes conducted in the plasma process chamber2218 form the plasma 2218A between the electrostatic chuck and a secondelectrode. As the plasma process is executed many aspects of the plasma2218A such as pressure, process gas mixtures and concentrations,temperatures, plasma byproduct content of the plasma, vary dynamically.As the various parameters of the plasma 2218A vary dynamically, theimpedance of the plasma also varies dynamically. Further, adding an RFprobe to the plasma 2218A changes the plasma rendering it very difficultto accurately measure the RF and other aspects of the plasma during thedynamic plasma process. As the impedance of the plasma 2218A varies, thevoltages and currents induced into the electrostatic chuck 2220 alsovary. As a result, it is very difficult to calculate a useful, accurateRF model of the electrostatic chuck 2220 using the above describedapproaches to modeling each stage of the RF transmission system 2200.

The RF return path 2290 is a complex path as the RF return signal iscoupled through many different structures and components within theplasma process chamber 2218 and external from the plasma process chamberto the return terminal of the RF generator 2202. As a result, the RFreturn path 2290 can also be difficult to model with equivalentrepresentative shunt and series RLC circuits. The RF return path 2290 isshown tied to ground potential 2291 as an example implementation onlyand it should be understood that the RF return path may or may not betied to a ground potential.

The stages 2230-2235 of the RF transmission system 2200 from the RFgenerator 2202 to the input to the electrostatic chuck 2220 can beaccurately modeled as described above. Unfortunately, the equivalentseries and shunt RLC circuit model of the entire RF transmission system2200, including the electrostatic chuck 2220, the plasma 2218A and theRF return path 2290 cannot be accurately modeled using the abovedescribed techniques.

FIG. 24 is a flowchart of the method operations 2400 for determining anRF transmission line model for the electrostatic chuck 2220, the plasma2218A and the RF return path 2290, in accordance with an embodimentdescribed in the present disclosure. Determining the RF transmissionline model for the electrostatic chuck 2220, the plasma 2218A and the RFreturn path 2290 will improve the accuracy of the RF transmission linemodel of the entire RF transmission system 2200.

In an operation 2405, a baseline RF transmission line model for the RFtransmission system stages 223-2235 is generated as described in moredetail above. One use of the RF transmission line model is to predictthe voltage of the RF in the plasma (Vrf), a current of the RF in theplasma (Irf) and power of the RF in the plasma (Prf) as these valuescannot be directly measured. The predicted Vrf, Irf and Prf values canbe used to predict a predicted RF induced DC bias voltage using the RFtransmission line model. During plasma processes, the actual RF inducedDC bias voltage can be measured on the surface of the wafer 2223 withvoltage probe 2221.

In an operation 2410, a predicted baseline Vrf, baseline Irf andbaseline Prf are determined using the baseline RF transmission linemodel and several input conditions. The predicted baseline Vrf, baselineIrf and baseline Prf are then used calculate a corresponding predictedbaseline RF induced DC bias voltage.

The predicted baseline RF induced DC bias voltage is then compared to anactually measured RF induced DC bias voltage. A baseline model errordefined as the differences between the predicted baseline RF induced DCbias voltage and the actually measured RF induced DC bias voltage. Inone example, the baseline model error can be as much as about 10percent. An accuracy of less than a 10 percent error rate is preferred.The error rate is caused in part by the baseline RF transmission linemodel not including accurate, equivalent series and shunt RLC circuitsfor the electrostatic chuck 2220, the plasma 2218A and the RF returnpath 2290.

In an operation 2415, an end module having the equivalent series andshunt RLC circuits corresponding to the electrostatic chuck 2220, theplasma 2218A and the RF return path 2290 is added to the baseline RFtransmission line model to create multiple, revised RF transmission linemodels. Adding the end module includes identifying RLC values for theequivalent series and shunt RLC circuits for the electrostatic chuck2220, the plasma 2218A and the RF return path 2290 as described in moredetail in FIGS. 25 and 26. Each of the revised RF transmission linemodels corresponds to one of the local minimums identified in FIGS. 25and 26.

In an operation 2430, the V′rf, I′rf and P′rf is calculated using eachof the revised RF transmission line models. In an operation 2435, theV′rf, I′rf and P′rf of each of the revised RF transmission line modelsis scored to identify a best fitting one of the revised RF transmissionline models as described in FIG. 27. The best fitting one of the revisedRF transmission line models is recorded in operation 2440 as a completedRF transmission line model and the method operations can end.

FIG. 25 is a flowchart of the method operations 2415 for adding an endmodule to the baseline RF transmission line model, in accordance with anembodiment described in the present disclosure. In an operation 2505,ranges for the initial RLC series and shunt values are selected. Theinitial ranges can be selected randomly or based on a best estimation ofthe ranges. By way of example, in reviewing the previously modeledstages 2230-2235 of the baseline RF transmission line model some rangesfor the RLC series and shunt values for each of the stages fell intoapproximated ranges. Specifically, for the series values: the Rservalues generally fell in a range of about 0 to about 50 ohms, the Lservalues generally fell in a range of about 0 to about 50,000 microhenriesand the Cser values generally fell in a ranges of about 0 to about50,000 picofarads. Similarly, for the shunt values: the Rsh valuesgenerally fell in a range of about 0 to about 50 ohms, the Lsh valuesgenerally fell in a range of about 0 to about 50,000 microhenries andthe Csh values generally fell in a range of about 0 to about 50,000picofarads. The estimated ranges of the RLC values noted in thepreviously modeled stages 2230-2235 were doubled to attempt to capturethe actual RLC values for the end module. Continuing the example: Theinitial ranges for the Rser and Rsh values were selected as 0-100 ohmsrange. The initial ranges for the Lser and Lsh were selected as 0 toabout 100,000 microhenries range. The initial ranges for the Cser andCsh were selected as 0 to about 100,000 picofarads range.

In an operation 2510, the ranges for the initial RLC series and shuntvalues are subdivided into a selected number of subdivisions. Any numberof subdivisions can be selected, however the greater the number ofsubdivisions, the greater the number of possible combinations of valueswould be tested. Given sufficient computing power, each of the ranges ofthe RLC values could be divided into single units such as there could be100,000 different values for Lser, Lsh, Cser and Csh and 100 differentvalues for each of Rser and Rsh, however, that would result in100,000*100,000*100,000*100,000*100*100=10²⁴ different combinations totest which would require a supercomputer to calculate within areasonable amount of time.

The ranges for the initial RLC series and shunt values are subdivided asfollows to reduce the number of combinations to a more workable number:10 different subdivisions for Lser, Lsh, Cser and Csh and 100 differentsubdivisions for each of Rser and Rsh, giving 100 million possiblecombinations to test. It should be understood that these are examplenumbers of subdivisions and more or fewer subdivisions could be chosenfor each of the ranges of each of the test RLC series and shunt values:Rser, Rsh, Lser, Lsh, Cser and Csh.

In an operation 2515, each combination of the test RLC series and shuntvalues is evaluated using a simplified bias model. The simplified biasmodel is also referred to as a linear bias model. The linear bias modelis a simplified version of a 79 term, 8^(th) order bias model. Thelinear bias model produces a linear relationship between the RF inducedDC bias voltage and Vrf or the Id. The linear bias model can beexpressed in voltage or current to calculate the corresponding valueincludes:

Voltage Model:a1*V2+a2*V27+a3*V60+a4*V2*V27+a5*V2*V60+a6*V27*V60+a7*V2*V27*V60+a8Current Model:a1*I2+a2*I27+a3*I60+a4*I2*I27+a5*I2*I60+a6*I27*I60+a7*I2*I27*I60+a8

Where: a1 through a8 are constant values derived using a least squaresregression. V2 is equal to the Vrf at 2 MHz. V27 is equal to the Vrf at27 MHz. V60 is equal to the Vrf at 60 MHz. I2 is equal to the Id at 2MHz. I27 is equal to the Irf at 27 MHz. I60 is equal to the Irf at 60MHz. It should be noted that the frequencies of 2, 27 and 60 are forpurposes of example explanation only and any one or more frequencies canbe used, where the frequencies are not limited to the range of 2-60 MHz.The voltage model and the current model can each be used to calculate acorresponding predicted RF induced DC bias voltage.

The Vrf output from the RF transmission line model is generally onlyaffected by the RLC series components of the final transmission linemodule and the Id output from the transmission line model is generallyonly affected by the RLC shunt components of the final transmission linemodule. Using the voltage and current models above, the series portionand the shunt portion can be optimized separately. This can also reducethe number of possible combinations between the series and shunt RLCvalues. It should be noted that while only two linear models were used,e.g., the voltage and the current models, any bias model could be usedwith this method and would produce a different set of local minimums.However, care should be taken when selecting the bias model for thismethod so as to avoid fitting the input data to the model chosen insteadof fitting the model to the data or to the inherent reality of thesystem.

Testing the test RLC values with the linear bias models may identifymultiple local minimums in goodness of fit to measured RF induced DCbias voltage in the RLC series or RLC shunt parameter space. Each of theidentified local minimums and the corresponding RLC series and shuntvalues are recorded such as being stored in a memory system such as adatabase or a table in an operation 2520.

A sufficient number of minimums are needed to increase the statisticallikelihood of capturing the actual global minimum as one of the recordedlocal minimums. The threshold number of local minimums can be selectedat any desired level. Continuing the above example, 1000 local minimumswere selected. In an operation 2525, the number of recorded localminimums is compared to the selected threshold number of local minimumsto determine if additional local minimums are needed.

If additional local minimums are needed in operation 2525, then themethod operations continue in an operation 2530 where additional rangesfor the RLC series and shunt values are selected and the methodoperations continue in operation 2510 as described above. If theselected threshold number of local minimums has been met or exceeded inoperation 2525 then the method operations continue in an operation 2430as discussed above in FIG. 24.

FIG. 26 is a flowchart of an alternate method operations 2415′ foradding an end module to the baseline RF transmission line model, inaccordance with an embodiment described in the present disclosure. In anoperation 2605, random test values for Rser, Lser, Cser, Rsh, Lsh andCsh are selected. In an operation 2610, the selected test values forRser, Lser, Cser, Rsh, Lsh and Csh are tested using the linear biasmodel as described above to identify a gradient with respect tocorresponding RF induced DC bias voltage goodness of fit. Identifying agradient is a mathematical analysis of comparing points on each side ofthe identified point, in each variable (Rser, Lser, Cser, Rsh, Lsh andCsh) and determining the direction of the gradient toward a localminimum.

In an operation 2615, the local gradient is followed toward the localminimum and new values for each of Rser, Lser, Cser, Rsh, Lsh and Cshare selected in the direction indicated by the gradient. By way ofexample, if the gradient indicates that the curve is sloped downward asRser is increased, then the new Rser value is selected in a slightlyincreased value from the previously selected value of Rser. Similarly,each of the remaining values for each of Rser, Lser, Cser, Rsh, Lsh andCsh are selected.

In one implementation, the gradient is identified by selecting one ofthe serial and shunt RLC values and then selecting a first gradient testvalue offset a selected positive increment from, e.g., is greater than,the selected serial and shunt RLC value. A second gradient test valuecan be selected. The second gradient test value is offset a selectednegative increment from, e.g., is less than, the selected serial andshunt RLC value. The gradient is calculated using the first gradienttest value and the second and the gradient test value and establishing aslope e.g., gradient, of a line passing through the first gradient testvalue and the second gradient test value.

If the gradient is negative, e.g., downward, from the second gradienttest value toward the first gradient test value, then a local minimum islikely located some unknown distance in a direction greater than thefirst gradient test value. Following the gradient, the first gradienttest value can be selected as a new second gradient test value and a newfirst gradient test value selected. The new first gradient test value isoffset a positive increment from the new second gradient test value andthe gradient through the new first gradient test value and the newsecond gradient test value is determined and the process repeatediteratively until a local minimum is found.

If the gradient is positive, e.g., upward, from the second gradient testvalue toward the first gradient test value, then a local minimum islikely located some unknown distance in a direction less than the secondgradient test value. Following the gradient, the second gradient testvalue can be selected as a new first gradient test value and a newsecond gradient test value selected. The new second gradient test valueis offset a negative increment from the new first gradient test valueand the gradient through the new first gradient test value and the newsecond gradient test value is determined and the process repeatediteratively until a local minimum is found.

In another implementation, the gradient is identified by selecting oneof the serial and shunt RLC values and then selecting a first gradienttest value offset a selected positive increment from, e.g., is greaterthan, the selected serial and shunt RLC value. Optionally, the firstgradient test value can be offset a selected negative increment from,e.g., is less than, the selected serial and shunt RLC value. Thegradient is calculated using the first gradient test value and theselected serial and shunt RLC value and establishing a slope e.g.,gradient, of a line passing through the first gradient test value andthe selected serial and shunt RLC value.

If the gradient is negative, e.g., downward, from the selected serialand shunt RLC value toward the first gradient test value, then a localminimum is likely located some unknown distance in a direction greaterthan the first gradient test value. Following the gradient, the firstgradient test value can be selected as a new second gradient test valueand a new first gradient test value selected. The new first gradienttest value is offset a positive increment from the new second gradienttest value and the gradient through the new first gradient test valueand the new second gradient test value is determined and the processrepeated iteratively until a local minimum is found.

If the gradient is positive, e.g., upward, from the selected serial andshunt RLC value test value toward the first gradient test value, then alocal minimum is likely located some unknown distance in a directionless than the second gradient test value. Following the gradient, thesecond gradient test value can be selected as a new first gradient testvalue and a new second gradient test value selected. The new secondgradient test value is offset a negative increment from the new firstgradient test value and the gradient through the new first gradient testvalue and the new second gradient test value is determined and theprocess repeated iteratively until a local minimum is found.

In another implementation, the gradient is identified by selecting oneof the serial and shunt RLC values and then selecting a first gradienttest value offset a selected negative increment from, e.g., is lessthan, the selected serial and shunt RLC value. The gradient iscalculated using the first gradient test value and the selected serialand shunt RLC value and establishing a slope e.g., gradient, of a linepassing through the first gradient test value and the selected serialand shunt RLC value.

If the gradient is negative, e.g., downward, from the first gradienttest value toward the selected serial and shunt RLC value, then a localminimum is likely located some unknown distance in a direction greaterthan the selected serial and shunt RLC value. Following the gradient,the selected serial and shunt RLC value can be selected as a new secondgradient test value and a new first gradient test value selected. Thenew first gradient test value is offset a positive increment from thenew second gradient test value and the gradient through the new firstgradient test value and the new second gradient test value is determinedand the process repeated iteratively until a local minimum is found.

If the gradient is positive, e.g., upward, from the first gradient testvalue toward the selected serial and shunt RLC value test value, then alocal minimum is likely located some unknown distance in a directionless than the first gradient test value. Following the gradient, thefirst gradient test value can be selected as a new first gradient testvalue and a new second gradient test value selected. The new secondgradient test value is offset a negative increment from the new firstgradient test value and the gradient through the new first gradient testvalue and the new second gradient test value is determined and theprocess repeated iteratively until a local minimum is found.

A local minimum is encountered when adjusted higher and lower in valuein a selected RLC test value indicates moving upward in the curve, awayfrom the current minimum. Each of the identified local minimums and thecorresponding RLC values are recorded such as being stored in a memorysystem such as a database or a table in an operation 2520.

A sufficient number of minimums are needed to increase the statisticallikelihood of capturing the actual global minimum as one of the recordedlocal minimums. The threshold number of local minimums can be selectedat any desired level. Continuing the above example, 1000 local minimumscan be selected. In an operation 2635, the number of recorded localminimums is compared to the selected threshold number of local minimumsto determine if the threshold has been met.

If, in operation 2635, the threshold has not been met, then the methodoperations continue in an operation 2605 where additional random testvalues for the Rser, Lser, Cser, Rsh, Lsh and Csh are selected. If theselected threshold number of local minimums has been met or exceeded inoperation 2635 then the method operations continue in an operation 2430as discussed above in FIG. 24.

FIG. 27 is a flowchart of the method operations 2435 for scoring each ofthe revised RF transmission line models, in accordance with anembodiment described in the present disclosure. In an operation 2705,one or more bias models are selected to test the accuracy of each of therevised RF transmission line models. By way of example, the linear biasmodel described above or the 79 term, 8^(th) order bias model or someother suitable bias model can be selected to evaluate each of therevised RF models. The RLC values of each of the revised RF transmissionline models are input to each of the selected bias model and thecorresponding predicted RF induced DC bias voltage are calculated in anoperation 2710 and stored in an operation 2715. Each of the bias modelscan produce a different predicted RF induced DC bias voltage value.

In an operation 2720, the predicted RF induced DC bias voltage arecompared to an empirically measured induced DC bias voltage. A score isassigned to each of the revised RF transmission line modelscorresponding to the accuracy of the predicted RF induced DC biasvoltage produced by the revised RF model. The score for each revised RFtransmission line models is recorded in an operation 2725 and the methodoperations continue in operation 2440 as discussed above.

It is also noted that the use of three parameters, e.g., RF voltage, RFcurrent, and the phase between the RF current and the RF voltage (or RFpower), etc., to model wafer bias allows better determination of waferbias voltage compared to the use of RF voltage alone. For example, waferbias calculated using the three parameters has a stronger correlation tonon-linear plasma regimes compared to a relation between RF voltage andthe non-linear plasma regimes. As another example, wafer bias calculatedusing the three parameters is more accurate than that determined using avoltage probe alone.

It is further noted that although the above-described operations aredescribed with reference to a parallel plate plasma chamber, e.g., acapacitively coupled plasma chamber, etc., in some embodiments, theabove-described operations apply to other types of plasma chambers,e.g., a plasma chamber including an inductively coupled plasma (ICP)reactor, a transformer coupled plasma (TCP) reactor, conductor tools,dielectric tools, a plasma chamber including an electron-cyclotronresonance (ECR) reactor, etc. For example, the x MHz RF generator andthe y MHz RF generator are coupled to an inductor within the ICP plasmachamber.

It is also noted that although the operations above are described asbeing performed by the processor of the host system 130 (FIG. 1), insome embodiments, the operations may be performed by one or moreprocessors of the host system 130 or by multiple processors of multiplehost systems.

It should be noted that although the above-described embodiments relateto providing an RF signal to the lower electrode of the ESC 177 (FIGS. 1and 18) and to the lower electrode of the ESC 192 (FIG. 11), andgrounding the upper electrodes 179 and 264 (FIGS. 1 and 11), in severalembodiments, the RF signal is provided to the upper electrodes 179 and264 while the lower electrodes of the ESCs 177 and 192 are grounded.

Embodiments described herein may be practiced with various computersystem configurations including hand-held hardware units, microprocessorsystems, microprocessor-based or programmable consumer electronics,minicomputers, mainframe computers and the like. The embodiments canalso be practiced in distributed computing environments where tasks areperformed by remote processing hardware units that are linked through anetwork.

With the above embodiments in mind, it should be understood that theembodiments can employ various computer-implemented operations involvingdata stored in computer systems. These operations are those requiringphysical manipulation of physical quantities. Any of the operationsdescribed herein that form part of the embodiments are useful machineoperations. The embodiments also relates to a hardware unit or anapparatus for performing these operations. The apparatus may bespecially constructed for a special purpose computer. When defined as aspecial purpose computer, the computer can also perform otherprocessing, program execution or routines that are not part of thespecial purpose, while still being capable of operating for the specialpurpose. In some embodiments, the operations may be processed by ageneral purpose computer selectively activated or configured by one ormore computer programs stored in the computer memory, cache, or obtainedover a network. When data is obtained over a network, the data may beprocessed by other computers on the network, e.g., a cloud of computingresources.

One or more embodiments can also be fabricated as computer-readable codeon a non-transitory computer-readable medium. The non-transitorycomputer-readable medium is any data storage hardware unit that canstore data, which can be thereafter be read by a computer system.Examples of the non-transitory computer-readable medium include harddrives, network attached storage (NAS), ROM, RAM, compact disc-ROMs(CD-ROMs), CD-recordables (CD-Rs), CD-rewritables (CD-RWs), magnetictapes and other optical and non-optical data storage hardware units. Thenon-transitory computer-readable medium can include computer-readabletangible medium distributed over a network-coupled computer system sothat the computer-readable code is stored and executed in a distributedfashion.

Although the method operations in the flowchart of FIG. 2, FIG. 13, FIG.15, FIG. 17 and FIGS. 24-27 above were described in a specific order, itshould be understood that other housekeeping operations may be performedin between operations, or operations may be adjusted so that they occurat slightly different times, or may be distributed in a system whichallows the occurrence of the processing operations at various intervalsassociated with the processing, as long as the processing of the overlayoperations are performed in the desired way.

One or more features from any embodiment may be combined with one ormore features of any other embodiment without departing from the scopedescribed in various embodiments described in the present disclosure.

Although the foregoing embodiments have been described in some detailfor purposes of clarity of understanding, it will be apparent thatcertain changes and modifications can be practiced within the scope ofthe appended claims. Accordingly, the present embodiments are to beconsidered as illustrative and not restrictive, and the embodiments arenot to be limited to the details given herein, but may be modifiedwithin the scope and equivalents of the appended claims.

1. Method for determining an RF transmission line model for a completeRF transmission system comprising: generating a baseline RF transmissionline model characterizing the RF transmission system, the baseline RFtransmission line model having a plurality of stages ending at an inputto an electrostatic chuck; calculating at least one of a plasma RFvoltage (Vrf), and/or a plasma RF current (Irf) and/or a plasma RF power(Prf) from the baseline RF transmission line model based on anempirically measured RF induced DC bias voltage; adding an end module tothe baseline RF transmission line model to create at least one revisedRF transmission line model, the end module including the electrostaticchuck, a plasma and an RF return path; calculating at least one of arevised plasma RF voltage (V′rf) and/or a revised plasma RF current(I′rf) and/or a revised plasma RF power (P′rf) and/or a correspondingrevised RF induced DC bias voltage from each of the at least one revisedbaseline RF transmission line models; scoring each one of the at leastone revised RF transmission line models to identify a best fittingrevised RF transmission line model; and recording the best fittingrevised RF transmission line model as a complete RF transmission linemodel.
 2. The method of claim 1, wherein adding an end module to thebaseline RF transmission line model includes: selecting an initial rangefor each of equivalent serial and shunt RLC values representing the endmodule; dividing the selected ranges into a selected number ofsubdivisions to identify test values corresponding to each subdivisionfor each serial and shunt RLC values; testing each combination of thetest values for each of the serial and shunt RLC values to identify aplurality of local minimums; recording each of the plurality of localminimums; and selecting additional ranges for each of equivalent serialand shunt RLC values representing the end module when a selectedthreshold number of local minimums is not met.
 3. The method of claim 2,wherein selecting the initial range for each of equivalent serial andshunt RLC values representing the end module includes selecting a rangeof equivalent serial and shunt RLC values including the equivalentserial and shunt RLC values in at least one other stage of the RFtransmission system.
 4. The method of claim 2, wherein selecting theinitial range for each of equivalent serial and shunt RLC valuesrepresenting the end module includes selecting random ranges of valuesfor each equivalent serial and shunt RLC values.
 5. The method of claim2, wherein dividing the selected ranges into the selected number ofsubdivisions to identify test values corresponding to each subdivisionfor each serial and shunt RLC values includes dividing each of theselected ranges into an equal number of subdivisions for each of theranges of the serial and shunt RLC values.
 6. The method of claim 2,wherein dividing the selected ranges into the selected number ofsubdivisions to identify test values corresponding to each subdivisionfor each serial and shunt RLC values includes dividing each of theselected ranges into non-equal numbers of subdivisions for each of theranges of the serial and shunt RLC values.
 7. The method of claim 2,wherein selecting additional ranges for each of equivalent serial andshunt RLC values representing the end module includes selecting amultiple of the initial range for at least one of each of equivalentserial and shunt RLC values representing the end module.
 8. The methodof claim 2, wherein selecting additional ranges for each of equivalentserial and shunt RLC values representing the end module includesselecting a larger range for at least one of each of equivalent serialand shunt RLC values representing the end module.
 9. The method of claim1, wherein adding an end module to the baseline RF transmission linemodel includes: selecting an initial random value for each of equivalentserial and shunt RLC values representing the end module; testing eachcombination of the test values for each of the serial and shunt RLCvalues to identify a gradient; select new test values for each of theserial and shunt RLC values corresponding to the gradient toward a localminimum; recording the local minimum when a local minimum is identified;and selecting additional random values for each of equivalent serial andshunt RLC values representing the end module when the threshold numberof local minimums is not met.
 10. The method of claim 9, wherein testingeach combination of the test values for each of the serial and shunt RLCvalues to identify the gradient includes for each one of the serial andshunt RLC values: selecting one of the serial and shunt RLC values;selecting a corresponding first gradient test value offset a selectedpositive increment from the selected one of the serial and shunt RLCvalues; selecting a corresponding second gradient test value offset aselected negative increment from the selected one of the serial andshunt RLC values; and calculating a gradient using the first gradienttest value and the second and the gradient test value.
 11. The method ofclaim 9, wherein testing each combination of the test values for each ofthe serial and shunt RLC values to identify the gradient includes foreach one of the serial and shunt RLC values: selecting one of the serialand shunt RLC values; selecting at least one of a corresponding firstgradient test value offset a selected positive increment from theselected one of the serial and shunt RLC values and/or a correspondingsecond gradient test value offset a selected negative increment from theselected one of the serial and shunt RLC values; and calculating agradient using at least two of: the first gradient test value and/or thesecond and the gradient test value and/or the selected one of the serialand shunt RLC values.
 12. The method of claim 1, wherein scoring therevised RF transmission line models to identify the best fitting revisedRF transmission line model includes: selecting at least one bias modelto test each revised RF transmission line model; testing each revised RFtransmission line model with each of the selected at least one biasmodel; storing test results for each of the selected at least one biasmodel for each of the revised RF transmission line model; and scoringthe stored test results according to accuracy of the revised plasma RFvoltage (V′rf) and/or a revised plasma RF current (I′rf) and/or arevised plasma RF power (P′rf) and/or a corresponding revised RF inducedDC bias voltage corresponding to each of the at least one revisedbaseline RF transmission line models.
 13. The method of claim 1, whereincalculating at least one of a plasma RF voltage (Vrf), and/or the plasmaRF current (Irf) and/or the plasma RF power (Prf) and/or thecorresponding RF induced DC bias voltage from the baseline RFtransmission line model includes calculating the corresponding RFinduced DC bias voltage from the baseline RF transmission line model;and wherein calculating at least one of the revised plasma RF voltage(V′rf) and/or the revised plasma RF current (I′rf) and/or the revisedplasma RF power (P′rf) and/or a corresponding revised RF induced DC biasvoltage from each of the at least one revised baseline RF transmissionline models includes calculating the corresponding revised RF induced DCbias voltage from each of the at least one revised baseline RFtransmission line models.
 14. The method of claim 13, wherein scoringeach one of the at least one revised RF transmission line model toidentify the best fitting revised RF transmission line model includescomparing the corresponding revised RF induced DC bias voltage for eachone of the at least one revised RF transmission line model to the RFinduced DC bias voltage from the baseline RF transmission line model,wherein the best fitting revised RF transmission line model has theleast difference between the corresponding revised RF induced DC biasvoltage and the empirically measured RF induced DC bias voltage. 15.Method for determining an RF transmission line model for a complete RFtransmission system comprising: generating a baseline RF transmissionline model characterizing the RF transmission system, the baseline RFtransmission line model having a plurality of stages ending at an inputto an electrostatic chuck; calculating at least one of a plasma RFvoltage (Vrf) and/or a plasma RF current (Irf) and/or a plasma RF power(Prf) and/or a corresponding RF induced DC bias voltage from thebaseline RF transmission line model; adding an end module to thebaseline RF transmission line model to create at least one revised RFmodel, the end module including the electrostatic chuck, a plasma and anRF return path including selecting an initial random value for each ofequivalent serial and shunt RLC values representing the end module;calculating at least one of a revised plasma RF voltage (V′rf) and/or arevised plasma RF current (I′rf) and/or a revised plasma RF power (P′rf)and/or a corresponding revised RF induced DC bias voltage from each ofthe at least one revised baseline RF models; scoring the revised RFtransmission line models to identify a best fitting revised RFtransmission line model including: selecting at least one bias model totest each revised RF transmission line model; testing each revised RFtransmission line model with each of the selected at least one biasmodel; storing test results for each of the selected at least one biasmodel for each of the revised RF model; and scoring the stored testresults according to accuracy of the revised plasma RF voltage (V′rf)and/or a revised plasma RF current (I′rf) and/or a revised plasma RFpower (P′rf) and/or a corresponding revised RF induced DC bias voltagecorresponding to each of the at least one revised baseline RF models;and recording the best fitting revised RF transmission line model as acomplete RF transmission line model.
 16. A plasma processing systemcomprising: a plasma processing chamber; an RF transmission systemcoupled to an RF input of the plasma processing chamber; an RF generatorhaving an output coupled to the RF transmission system; and a controllercoupled to the RF generator and the plasma processing chamber, thecontroller including logic on computer readable media being executablefor adjusting at least one dynamic first principal variable of an RFsignal in a plasma in the plasma processing chamber according to acomplete RF transmission line model of the RF transmission system tocompensate for a difference between the plasma system and a secondplasma system, the complete RF transmission line model being determinedby: generating a baseline RF transmission line model characterizing theRF transmission system, the baseline RF transmission line model having aplurality of stages ending at an input to an electrostatic chuck;calculating at least one of a plasma RF voltage (Vrf) and/or a plasma RFcurrent (Irf) and/or a plasma RF power (Prf) and/or a corresponding RFinduced DC bias voltage from the baseline RF transmission line model;adding an end module to the baseline RF transmission line model tocreate at least one revised RF transmission line model, the end moduleincluding the electrostatic chuck, a plasma and an RF return path;calculating at least one of a revised plasma RF voltage (V′rf) and/or arevised plasma RF current (I′rf) and/or a revised plasma RF power (P′rf)and/or corresponding revised RF induced DC bias voltage from each of theat least one revised baseline RF transmission line models; scoring therevised RF transmission line models to identify a best fitting revisedRF transmission line model; and recording the best fitting revised RFtransmission line model as a complete RF transmission line model. 17.The system of claim 16, wherein adding an end module to the baseline RFtransmission line model includes: selecting an initial range for each ofequivalent serial and shunt RLC values representing the end module;dividing the selected ranges into a selected number of subdivisions toidentify test values corresponding to each subdivision for each serialand shunt RLC values; testing each combination of the test values foreach of the serial and shunt RLC values to identify a plurality of localminimums; recording each of the plurality of local minimums; andselecting additional ranges for each of equivalent serial and shunt RLCvalues representing the end module when the threshold number of localminimums is not met.
 18. The system of claim 16, wherein adding an endmodule to the baseline RF transmission line model includes: selecting aninitial random value for each of equivalent serial and shunt RLC valuesrepresenting the end module; testing each combination of the test valuesfor each of the serial and shunt RLC values to identify a gradient;select new test values for each of the serial and shunt RLC valuescorresponding to the gradient toward a local minimum; recording thelocal minimum when a local minimum is identified; and selectingadditional random values for each of equivalent serial and shunt RLCvalues representing the end module when the threshold number of localminimums is not met.
 19. The system of claim 16, wherein scoring therevised RF transmission line models to identify the best fitting revisedRF transmission line model includes: selecting at least one bias modelto test each revised RF transmission line model; testing each revised RFtransmission line model with each of the selected at least one biasmodel; storing test results for each of the selected at least one biasmodel for each of the revised RF transmission line model; and scoringthe stored test results according to accuracy of the revised plasma RFvoltage (V′rf), a revised plasma RF current (I′rf) and/or a revisedplasma RF power (Prf) corresponding to each of the at least one revisedbaseline RF transmission line models.